1. Introduction

Unless stated otherwise, graphs will be finite, undirected and connected simple graphs. A shortest path having end vertices \(u\) and \(v\) is denoted by, \(u-v_{(in~G)}\). If \(d_G(u,v)\geq 2\) then a vertex \(w\) on \(u-v_{(in~G)}\), \(w\neq u\), \(w\neq v\) is called an internal vertex on \(u-v_{(in~G)}\). When the context is clear the notation such as \(d_G(u,v)\), \(deg_G(v)\) can be abbreviated to \(d(u,v)\), \(deg(v)\) and so on. Good references to important concepts, notation and graph parameters can be found in [1,2,3].

The notions of parametric equivalence, isomorphism and uniqueness had been introduced in [4]. For ease of reference we recall from [4] as follows: Let \(\rho\) denote some minimum or maximum graph parameter related to subsets \(V(G)\) of graph \(G\). Vertex subsets \(X\) and \(Y\) is said to be parametric equivalent or \(\rho\)-equivalent if and only if both \(X\), \(Y\) satisfy the parametric conditions of \(\rho\). This relation is denoted by \(X\equiv_\rho Y\). Furthermore, if \(X\equiv_\rho Y\) and the induced graphs \(\langle V(G)\backslash X\rangle \cong \langle V(G)\backslash Y\rangle\) then \(X\) and \(Y\) are said to be parametric isomorphic. This isomorphic relation is denoted by \(X \cong_\rho Y\). Let all possible vertex subsets of graph \(G\) which satisfy \(\rho\) be \(X_1, X_2, X_3,\dots, X_k\). If \(X_1 \cong_\rho X_2 \cong_\rho X_3 \cong_\rho \cdots \cong_\rho X_k\) then \(X_i\), \(1\leq i \leq k\) are said to be parametric unique or \(\rho\)-unique. The graph \(G\) is said to have a parametric unique or \(\rho\)-unique solution (or parametric unique \(\rho\)-set). If \(G\) has a unique (exactly one) \(\rho\)-set \(X\), then \(X\) is a parametric unique \(\rho\)-set.

This paper furthers the introductory research presented in [4].

2. Confluence in graphs

Shiny et al., [5] introduced the concept of a confluence set (a subset of vertices) of a graph \(G\), also see [6] for results on certain derivative graphs. Recall that for a non-complete graph \(G\), a non-empty subset \(\mathcal{X}\subseteq V(G)\) is said to be a confluence set if for every unordered pair \(\{u,v\}\) of distinct vertices (if such exist) in \(V(G)\backslash \mathcal{X}\) for which \(d_G(u,v)\geq 2\) there exists at least one \(u-v_{(in~G)}\) with at least one internal vertex, \(w\in \mathcal{X}\). Also a vertex \(u\in \mathcal{X}\) is called a confluence vertex of \(G\). A minimal confluence set \(\mathcal{X}\) (also called a \(\zeta\)-set) has no proper subset which is a confluence set of \(G\). The cardinality of a minimum confluence set is called the confluence number of \(G\) and is denoted by \(\zeta(G)\). A minimal confluence set is denoted by \(\mathcal{C}\). To distinguish between different graphs the notation \(\mathcal{C}_G\) may be used for a minimum confluence set of \(G\). We recall two important results from [4]. We remind that for a complete graph the confluence number is \(0\) hence, \(\mathcal{C}_{K_n}=\emptyset\), \(n\geq 1\).

Proposition 1. [4] A path \(P_n\) has a parametric unique \(\zeta\)-set if and only if \(n=1,2\) or \(n=4+3i\) or \(n=5+3i\), \(i=0,1,2,\dots\).

Proposition 2. [4] A cycle \(C_n\) has a parametric unique \(\zeta\)-set if and only if \(n=3,4\) or \(n=5+3i\) or \(n=6+3i\), \(i=0,1,2,\dots\).

2.1. Cycle related graphs

Henceforth, a cycle \(C_n\), \(n\geq 3\) of order \(n\) has the vertex set \(V(C_n)=\{v_i:i=1,2,3,\dots,n\}\).

• (a)   A wheel graph (simply, a wheel) \(W_n\) is obtained from a cycle \(C_n\), \(n\geq 3\) with an additional central vertex \(v_0\) and the additional edges \(v_0v_1\), \(1\leq i\leq n\). The cycle is called the rim and the edges \(v_0v_i\), \(1\leq i\leq n\) are called spokes. Alternatively, \(W_n=C_n+K_1\) and \(V(K_1)=\{v_0\}\).

Proposition 3. A wheel graph \(W_n\) has a parametric unique \(\zeta\)-set.

Proof. Since \(W_3\) is complete the result is trivial. For \(n\geq 4\) the distance \(d(v_i,v_j)\leq 2\) for all distinct pairs. For \(i,j\neq 0\) and \(v_i\) not adjacent to \(v_j\) there exists a \(3\)-path (or \(2\)-distance path) with \(v_0\) the internal vertex. Hence, the unique \(\zeta\)-set is \(\{v_0\}\), therefore parametric unique.

• (b)   A helm graph \(H_n\) is obtained from a wheel graph \(W_n\) by adding a pendent vertex (or leaf) \(u_i\) to each rim vertex \(v_i\).

Proposition 4.

• (a)   The helm graph \(H_3\) does not have a parametric unique \(\zeta\)-set.
• (b)   A helm graph \(H_n\), \(n\geq 4\) has a parametric unique \(\zeta\)-set.

Proof.

• (a)   Consider \(H_3\). Clearly and without loss of generality the sets \(X_1=\{v_0,v_1,v_2\}\), \(X_2 = \{v_1,v_2,v_3\}\) and \(X_3=\{v_1,v_2,u_3\}\) are all minimal confluence sets. Hence \(\zeta(H_3)\leq 3\). It is easy to verify that no \(2\)-vertex subset is a confluence set. Thus, \(\zeta(H_3)>2\). Also, \(\langle V(H_3)\backslash X_1\rangle \ncong \langle V(H_3)\backslash X_2\rangle\). Therefore \(H_3\) does not have a parametric unique \(\zeta\)-set. The aforesaid follows in essence from the fact that \(H_3\) is complete. Therefore, it is not necessary for \(v_0\) to be in all \(\zeta\)-sets.
• (b)   For \(H_n\), \(n\geq 4\) the distance \(d(u_i,u_{i+1})=3\) hence a rim vertex is required. The distance \(d(u_i,u_{i+2})=5\) hence the vertex \(v_0\) will suffice along the \(5\)-path \(u_iv_iv_0v_{i+2}u_{i+2}\). By symmetry considerations and therefore up to isomorphism and without loss of generality we have two subcases.

  Subcase 1. If \(n\) is even the set \(X_1=\{v_0, v_1,v_3,v_5,\dots,v_{n-1}\}\) is a \(\zeta\)-set and clearly \(H_n\) has a parametric unique \(\zeta\)-set.

  Subcase 2. If \(n\) is odd the sets \(X_1=\{v_0, v_1,v_3,v_5,\dots,v_{n-1}\}\) and \(X_2=\{v_0, v_1,v_3,v_5,\dots,v_{n-2},v_n\}\) are a \(\zeta\)-sets. Clearly \(\langle V(H_n)\backslash X_1\rangle \cong \langle V(H_n)\backslash X_1\rangle\). Thus \(H_n\) has a parametric unique \(\zeta\)-set.

As a direct consequence of the proof of Proposition 4, we get the next corollary.

Corollary 1. A helm graph has \(\zeta(H_n)=\lceil \frac{n}{2}\rceil +1\).

• (c)   A flower graph \(Fl_n\) is obtained from a helm graph \(H_n\) by adding the edges \(v_0u_i\), \(1\leq i\leq n\).

Proposition 5. A flower graph \(Fl_n\) has a parametric unique \(\zeta\)-set.

Proof. The result follows by similar reasoning as in the proof of Proposition 3.

As a direct consequence of Proposition 5, we get the next corollary.

Corollary 2. A flower graph has \(\zeta(Fl_n)=1\).

• (d)   A closed helm graph \(H^c_n\) is obtained from a helm graph \(H_n\) by completing a cycle, \(C'_n=u_1u_2u_3\cdots u_nu_1\) on the leafs of \(H_n\).

Proposition 6.

• (a)   A closed helm graph \(H^c_n\) for \(n=4\) or \(n\) is odd does not have a parametric unique \(\zeta\)-set.
• (b)   A closed helm graph \(H^c_n\), \(n \geq 6\) and even, has a parametric unique \(\zeta\)-set.

Proof. It is easy to verify that all distance paths such that \(d(u_i,u_j)\leq 3\) are paths on \(C'_n\). Also, for \(u_i,u_j\in \mathcal{C}_{C'_n}\) we have \(d(u_i,u_j)\leq 3\). It follows that \(\mathcal{C}_{C'_n}\subseteq \mathcal{C}_{H^c_n}\).

• (a)   By similar reasoning to that in the proof of Proposition 4(a) it follows that \(H^c_3\) and \(H^c_4\) do not have a unique \(\zeta\)-set.

  From the set \(X_1=\{v_i: u_i\notin \mathcal{C}_{C'_n}\}\cup \{v_0\}\) it is possible to select a minimum confluence set in respect of the spanning subgraph \(H_n\) say set \(X_2\). The set \(\mathcal{C}_{H^c_n}=\mathcal{C}_{C'_n}\cup X_2\) is a minimum confluence set.

  Subcase (a)(1). Since by symmetry the choice of say, \(X_2\) can be fixed, For \(n\geq 5\) and odd, the choice of \(\mathcal{C}_{C'_n}\) can rotate such that \(\langle V(H^c_n)\backslash \mathcal{C}_{H^c_n}\rangle\) does not remain isomorphic.

• (b)   By similar reasoning \(X_2\) can be fixed. However, for \(n\geq 6\) and even and by symmetry properties of \(C'_n\) all choices of \(\mathcal{C}_{C'_n}\) yield isomorphic \(\langle V(H^c_n)\backslash \mathcal{C}_{H^c_n}\rangle\).

As a direct consequence of the proof of Proposition 6, we get the next corollary.

Corollary 3. A closed helm graph has \(\zeta(H^c_n)= \lceil \frac{n}{2}\rceil +1\).

• (e)   A gear graph \(G_n\) is obtain from a wheel graph \(W_n\) by inserting a vertex \(u_i\) on the edge \(v_iv_{i+1}\) and \(n+1\equiv 1\). Note that \(G_n\) has \(2n+1\) vertices and \(3n\) edges. The rim is now called a boundary cycle denoted by \(C^b(G_n)\).

Proposition 7.

• (a)   \(G_3\) has a parametric unique \(\zeta\)-set.
• (b)   A gear graph \(G_n\) and \(n \geq 5\) is odd does not have a parametric unique \(\zeta\)-set.
• (c)   A gear graph \(G_n\) and \(n \geq 4\) is even has a parametric unique \(\zeta\)-set.

Proof.

• (a)   For \(G_3\) it follows easily that up to isomorphism the \(\zeta\)-set \(\{u_1,v_3\}\) is unique.
• (b)   The inner-area enclosed by the cycle \(C'_{2n} =v_1u_1v_2u_2\cdots v_nu_nv_1\) can be partitioned into \(n\) planar areas, each enclosed by a \(C_4\). For all pairs \(v_i,v_j\) it is necessary and sufficient that \(v_0\in \zeta\)-set. Let \(n \geq 5\) be odd. Without loss of generality, an optimal minimal confluence set is given by \(X_1=\{v_0,u_1,u_3,\dots, u_{n-2},u_{n-1}\}\) or \(X_2=\{v_0,u_1,u_3,\dots, u_{n-2},v_n\}\) or \(X_3=\{v_0,u_1,u_3,\dots, u_{n-2},u_n\}\). Hence, \(\zeta(G_n)\leq \lceil \frac{2n}{4}\rceil +1= \lceil \frac{n}{2}\rceil +1\). Because the boundary cycle \(C^b(G_n)\) has \(\zeta(C^b(G_n))= \lceil \frac{2n}{3}\rceil\) it follows that \(\zeta(G_n) \geq \lceil \frac{2n}{3}\rceil\). However for \(n\) is odd, \(\lceil \frac{2n}{3}\rceil = \lceil \frac{n}{2}\rceil +1\). Since, \[\langle V(G_n)\backslash X_1\rangle \ncong \langle V(G_n)\backslash X_2\rangle.\] It follows that a gear graph \(G_n\) does not have a parametric unique \(\zeta\)-set for \(n\) is odd.
• (c)   For \(n \geq 4\) and even, reasoning similar to that in (b) show that up to isomorphism the \(\zeta\)-set \(X_1=\{v_0,u_1,u_3,\dots, u_{n-2},u_{n-1}\}\) is unique. Reasoning in respect of bounds on \(\zeta(G_n)\) similar to that in (a) settles the result.

As a direct consequence of the proof of Proposition 7, we get the next corollary.

Corollary 4. The gear graph \(G_3\) has \(\zeta(G_3)=2\). A gear graph of order \(n\geq 4\) has \(\zeta(G_n)=\lceil \frac{n}{2}\rceil +1\).

• (f)   A sun graph \(S^\boxtimes_n\), \(n\geq 3\) is obtained by taking the complete graph \(K_n\) on the vertices \(v_1,v_2,v_3,\dots,v_n\) together the isolated vertices \(u_i\), \(1\leq i\leq n\) and adding the edges \(v_iu_i\), \(u_iv_{i+1}\) and \(n+1\equiv 1\). The boundary cycle of a sun graph is the cycle \(C^b(S^\boxtimes_n) =v_1u_1v_2u_2v_3u_3\cdots u_nv_1\).

Proposition 8. A sun graph \(S^\boxtimes_n\), \(n\geq 3\) has a parametric unique \(\zeta\)-set if and only if \(C^b(S^\boxtimes_n)\) is of order \(n= 3i\), \(i=1,2,3,\dots\)

Proof. Since all pairs \(v_i,v_j\) are adjacent it suffices to only consider a \(\zeta\)-set of \(C^b(S^\boxtimes_n)\). Since \(deg(u_i)=2\) and \(deg(v_j)=3\) any \(\zeta\)-set must be graphically symmetrical for a sun graph to have a parametric unique \(\zeta\)-set. A graphically symmetrical \(\zeta\)-set means that, measured along the boundary cycle, \(min\{d(v_j,u_k):v_j,u_k\in \zeta\)-set\(\}=3\). It implies that \(n=3i\), \(i=1,2,3,\dots\).

The converse follows from the fact that sun graphs with \(C^b(S^\boxtimes_n)\) of order \(n\neq 3i\), \(i=1,2,3,\dots\) do not have graphically symmetrical \(\zeta\)-sets of even order.

Note that if a sun graph has a parametric unique \(\zeta\)-set then \(\zeta(S^\boxtimes_n)\) is even. Furthermore, as a direct consequence of the proof of Proposition 8, we get the next corollary.

Corollary 5. A sun graph has \(\zeta(S^\boxtimes_n)=\lceil \frac{2n}{3}\rceil\).

• (g)   A sunflower graph \(S^\circledast_n\), \(n\geq 3\) is obtained by taking the wheel graph \(W_n\) together the isolated vertices \(u_i\), \(1\leq i\leq n\) and adding the edges \(v_iu_i\), \(u_iv_{i+1}\) and \(n+1\equiv 1\). The boundary cycle of a sun graph is the cycle \(C^b(S^\circledast_n) =v_1u_1v_2u_2v_3u_3\cdots u_nv_1\).

Proposition 9. A sunflower graph \(S^\circledast_n\), \(n\geq 3\) does not have a parametric unique \(\zeta\)-set.

Proof. For all pairs \(v_i,v_j\) it is sufficient that \(v_0\in \zeta\)-set. Thereafter any \(\zeta\)-set \(X_1\) in respect of \(C^b(S^\circledast_n)\) is required to obtain \(\mathcal{C}_{S^\circledast_n}= X_1\cup \{v_0\}\). It implies that \(\zeta(S^\circledast_n)= n\). In turn, the aforesaid confluence number permits that say, \(X_2=\{v_1,v_2,v_3,\dots,v_n\}\) or \(X_3=\{v_1,v_2,v_3,\dots,v_{n-1},u_{n-1}\}\) are \(\zeta\)-sets. Since, \(\langle V(S^\circledast_n)\backslash X_1\rangle \ncong \langle V(S^\circledast_n)\backslash X_2\rangle \ncong \langle V(S^\circledast_n)\backslash X_3\rangle\) the result follows.

As a direct consequence of the proof of Proposition 9, we get the next corollary.

Corollary 6. A sunflower graph has \(\zeta(S^\circledast_n)= n\).

• (h)   A sunlet graph \(S^\circleddash_n\), \(n\geq 3\) is obtained by taking cycle \(C_n\) together the isolated vertices \(u_i\), \(1\leq i\leq n\) and adding the pendent edges \(v_iu_i\).

Proposition 10. A sunlet graph \(S^\circleddash_n\), \(n\geq 3\) has a parametric unique \(\zeta\)-set.

Proof. Case 1. Let \(n\geq 3\) and odd. Without loss of generality and by isomorphism, it is easy to verify that the sets \(X_1=\{v_1,v_3,v_5,\dots,v_n\}\) and \(X_2=\{v_1,v_3,v_5,\dots,v_{n-2},v_{n-1}\}\) are \(\zeta\)-sets. Furthermore, up to isomorphism those are the only distinguishable \(\zeta\)-sets. Since, \[\langle (V(S^\circleddash_n)\backslash X_1\rangle \cong \langle (V(S^\circleddash_n)\backslash X_2\rangle,\] the result follows for \(n\geq 3\) and odd.

Case 2. By similar reasoning as in Case 1 the result follows for \(n\geq 4\) and even.

As a direct consequence of the proof of Proposition 10, we get the next corollary.

Corollary 7. A sunlet graph has \(\zeta(S^\circleddash_n)=\lceil \frac{n}{2}\rceil\).

• (i)   A circular ladder (or prism graph) \(L^\circ_n\), \(n\geq 3\) is obtained by taking two cycles of equal order \(n\). Label as, \(C^1_n=v_1v_2v_3\cdots v_nv_1\) and \(C^2_n= u_1u_2u_3\cdots u_nu_1\). Add the edges \(v_iu_i\), \(1\leq i \leq n\). A circular ladder can be viewed as \(H^c_n-v_0\).

Proposition 11. A circular ladder graph \(L^\circ_n\) has a parametric unique \(\zeta\)-set if and only if \(n=4 \) or \(n=3i\) for \(i=2,3,4,...\).

Proof. Part 1. For \(n=4\), \(X_{i}=\{u_{i},v_{j}\}\), \(i=1,2,3,4\), \(j\in \{1,2,3,4\}\) such that \(d(u_{i},v_{j})=3\), are the minimum confluence sets for \(L^\circ_4\). Since \(\langle V(L^\circ_4)\backslash X_{i}\rangle\) are \(C_{6}\) for \(i=1,2,3,4\), we have the result for \(n=4\).

In a circular ladder graph \(L^\circ_n\), \(n\neq 4\) there are \(n\) copies of \(C_{4}=v_{i}u_{i}u_{i+1}v_{i+1}\). For each \(C_{4}=v_{i}u_{i}u_{i+1}v_{i+1}\), at least one of the vertices \(v_{i},u_{i},u_{i+1},v_{i+1}\) belongs to every minimum confluence set of \(L^\circ_n\).

Part 2. For \(n=3\), \(X_{1}=\{v_{1},v_{2}\}\) and \(X_{2}=\{v_{1},u_{2}\}\) are two minimum confluence set for \(L^\circ_3\). However, \(\langle V(L^\circ_3)\backslash X_{1}\rangle\) and \(\langle V(L^\circ_3) \backslash X_{2}\rangle\) are not isomorphic. Hence \(L^\circ_3\) has no unique parametric set.

Part 3. For \(n=3i\), \(i=2,3,..\), let \(\mathcal{C}_{C_{n}}(v_{i})\) be a minimum confluence set of \(C_{n}\) starting from \(v_{i}\) and \(C_{C_{n}^{'}}(u_{j})\) be a minimum confluence set of \(C_{n}^{'}\) starting from \(u_{j}\). Then for \(i \neq j\),\(X_{ij}= \mathcal{C}_{C_{n}}(v_{i}) \cup \mathcal{C}_{C_{n}^{'}}(u_{j})\) is a minimum confluence set for \(L^\circ_n\) and \(\langle V(L^\circ_n)\backslash X_{ij}\rangle\) consists of \(\frac{n}{3}\) copies of \(P_{3}\). Hence the result for \(n=3i\), \(i=2,3,..\).

Part 4. If \(n \equiv 2(mod~3)\). Let \(X_{1}\) be the minimum confluence set for \(L^\circ_n\) such that \(u_{i},u_{i+2},v_{i+1} \in X_{1}\) and let \(X_{2}\) be the minimum confluence set for \(L^\circ_n\) such that \(u_{i},u_{i+2},v_{i} \in X_{2}\). Then \(\langle V(L^\circ_n)\backslash X_{1}\rangle\) and \(\langle V(L^\circ_n)\backslash X_{2}\rangle\) are not isomorphic. Hence \(L^\circ_n\) has no parametric unique set if \(n_{\geq 5} \equiv 2(mod~3)\).

By a similar argument we have to prove that \(L^\circ_n\) has no parametric unique set if \(n_{\geq 7} \equiv 1(mod~3)\).

Since all \(n\in \mathbb{N}_{\geq 3}\) have been accounted for the 'if' has been settled.

For all valid cases the converse, 'only if', follows through reasoning by contradiction.

Corollary 8. A circular ladder has, \begin{equation*} \zeta(L^\circ_n) = \begin{cases} 2, & \mbox{if \(n= 4\)};\\ 2\lceil \frac{n}{3}\rceil, & \mbox{if \(n=3\) or \(n \geq 5\)}.\\ \end{cases} \end{equation*}

Proof. The result is a consequence of the proof of Proposition 11. The exception lies in the fact that \(L^\circ_4\) has \(5=n_{=4}+1\) cycles \(C_4\) to account for. All other \(L^\circ_{n_{\neq 4}}\) have \(n\) cycles \(C_4\) to account for.

Observe that the confluence number of a circular ladder is always even.

• (j)   A tadpole graph \(T(m,n)\), \(m\geq 3\), \(n\geq 1\) is obtained from a cycle \(C_m=v_1v_2v_3\cdots v_mv_1\) and a path \(P_n = u_1u_2u_3\cdots u_n\) by adding an edge between an end-vertex of \(P_n\) and a vertex of \(C_m\). The new edge is also called a bridge.

Proposition 12. A tadpole graph \(T(m,n)\), \(m\geq 3\), \(n\geq 1\):

• (a)   Tadpole graphs \(T(3,n)\), \(n\geq 1\) have a parametric unique \(\zeta\)-set if and only if \(n=3i\), \(i=1,2,3,\dots\).
• (b)   Tadpole graphs \(T(4,1)\), \(T(4,2)\) have a parametric unique \(\zeta\)-sets.
• (c)   Tadpole graphs \(T(5,1)\) does not have a parametric unique \(\zeta\)-set and \(T(5,2)\) has.
• (d)   Tadpole graphs \(T(m,1)\), \(T(m,2)\), \(m\geq 6\) have a parametric unique \(\zeta\)-set if and only if \(m=6+3i\), \(i= 0,1,2,\dots\)
• (e)   Tadpole graphs \(T(m,n)\), \(m\geq 4\) and \(n\geq 3\) have a parametric unique \(\zeta\)-set if and only if both the cycle \(C_m\) and the path \(P_n\) have parametric unique \(\zeta\)-sets.
• (f)   All other tadpole graphs as excluded through (a) to (f) do not have a parametric unique \(\zeta\)-set.

Proof.

• (a)   The tadpole graphs \(T(3,n)\), \(n\geq 1\) does not have a parametric unique \(\zeta\)-set for \(P_1\), \(P_2\) (straightforward).

  Subcase (a)(1). For \(n+2=5+3i\), \(i=0,1,2,\dots\) the \(\zeta\)-set of \(P_{n+2}\) is unique hence, \(T(3,n)\) has a parametric unique \(\zeta\)-set.

  Subcase (a)(2). For \(n+2=6+3i\), \(i= 0,1,2,\dots\) the \(\zeta\)-set of \(P_{n+2}\) is not parametric unique hence, \(T(3,n)\) does not have a parametric unique \(\zeta\)-set.

  Subcase (a)(3). For \(n+2=7+3i\), \(i= 0,1,2,\dots\) the \(\zeta\)-set of \(P_{n+2}\) is parametric unique. However, since some \(\zeta\)-sets may contain vertex \(v_j\) of the bridge the tadpole \(T(3,n)\) does not have a parametric unique \(\zeta\)-set.

  All tadpoles \(T(3,n)\), \(n\geq 1\) have been accounted for because, \[ \mathbb{N}=\{1,2\}\cup \{3+3i:i=0,1,2,\dots\}\cup \{4+3i:i=0,1,2,\dots\}\cup \{5+3i:i=0,1,2,\dots\}. \]
• (b)   The tadpole graphs \(T(4,n)\), \(n\geq 1\) have a parametric unique \(\zeta\)-set for \(P_1\), \(P_2\). It follows from the fact that a bridge vertex say, \(v_i\) has to be in any \(\zeta\)-set.

  Subcases \(n+2=5+3i\), \(n+2=6+3i\) and \(n+2=7+3i\), \(i= 0,1,2,\dots\) will be settled in (d) and (e) below.

• (c)   The tadpole graphs \(T(5,n)\), \(n\geq 1\) does not have a parametric unique \(\zeta\)-set for \(P_1\) bacause it is easy to verify that an end-vertex of the bridge need not be in all \(\zeta\)-sets. However for \(P_2\) the tadpole has a parametric unique \(\zeta\)-set. It follows from the fact that a bridge vertex say, \(v_i\) has to be in any \(\zeta\)-set.

  Subcases \(n+2=5+3i\), \(n+2=6+3i\) and \(n+2=7+3i\), \(i= 0,1,2,\dots\) will be settled in (d) and (e) below.

• (d)   The tadpoles \(T(m,1)\), \(T(m,2)\), \(m\geq 6\) do not require that vertices \(u_1\) and/or \(u_2\) to necessarily be in a \(\zeta\)-set. Hence, all \(\zeta\)-sets of cycle \(C_m\) which contain a vertex of the bridge suffice to be \(\zeta\)-sets of the tadpoles. Therefore has a parametric unique \(\zeta\)-set if and only if \(C_m\) has a unique \(\zeta\)-set. Therefore, if and only if \(m=6+3i\), \(i= 0,1,2,\dots\) The converse follows easily by contradiction.
• (e)   Finally, for a tadpole \(T(m,n)\), \(m\geq 4\) and \(n\geq 3\) and both the cycle \(C_m\) and the path \(P_n\) have parametric unique \(\zeta\)-sets, it is easy to verify that the \(\zeta\)-sets of the tadpole all contain a vertex \(v_j\) of the bridge. Therefore the tadpole has a parametric \(\zeta\)-set. Else, it is always possible to find a \(\zeta\)-set of the tadpole which contains a vertex \(v_j\) which is on the bridge and another \(\zeta\)-set which does not. Therefore, such tadpoles do not have a parametric unique \(\zeta\)-set. Hence, the tadpoles \(T(m,n)\), \(m\geq 4\) and \(n\geq 3\) have a parametric unique \(\zeta\)-set if and only if both \(C_m\) and \(P_n\) have parametric unique \(\zeta\)-sets.
• (f)   All other tadpole graphs which were excluded through reasoning of proof, (a) to (e) do not have a parametric unique \(\zeta\)-set.

• (k)   A lollipop graph \(L^\boxtimes(m,n)\), \(m\geq 3\), \(n\geq 1\) is obtained from a complete graph \(K_m\) and a path \(P_n\) by adding a bridge between an end-vertex of \(P_n\) and a vertex of \(C_m\).

Proposition 13. A lollipop graph \(L^\boxtimes(m,n)\), \(m\geq 3\), \(n\geq 1\) has a parametric unique \(\zeta\)-set if and only if \(n=3i\), \(i=1,2,3,\dots\).

Proof. The proof follows directly from the proof of Proposition 12(a).

• (l)   A generalized barbell graph \(B(n,m)\), \(n,m\geq 3\) is obtained from two complete graph \(K_n\), \(K_m\) and adding a bridge.

Proposition 14. A generalized barbell graph \(B(n,m)\), \(n,m\geq 3\) has a parametric unique \(\zeta\)-set if and only if \(n=m\).

Proof. Let \(K_n\) be on vertices \(v_1,v_2,v_3,\dots,v_n\) and \(K_m\) on vertices \(u_1,u_2,u_3,\dots,u_m\). For any pair \(v_iu_j\) and edge \(v_iu_j\) not the bridge, the distance \(d(v_i,u_j)=2\) or \(3\). Therefore any vertex of the bridge yields a \(\zeta\)-set. Without loss of generality let the \(\zeta\)-set be \(\{v_k\}\). It follows that \(\langle V(B(n,m))\backslash \{v_k\}\rangle \cong K_{n-1}\cup K_m\). Hence, \(B(n,m)\) has a parametric unique \(\zeta\)-set if and only if \(n=m\).

3. Conclusion

The study of cycle related graphs has not exhausted. Note that for those cycle related graphs which do not have a parametric unique \(\zeta\)-set the proof by contradiction can be utilized well.

The idea of combined parametric conditions remains open. Note that the parametric conditions will be ordered pairs. For example, the path \(P_3 = v_1v_2v_3\) has a unique minimum dominating set i.e. the \(\gamma\)-set \(X_1=\{v_2\}\). Since \(X_1\) is also a \(\zeta\)-set of \(P_3\) the set is said to be a parametric unique \((\gamma,\zeta)\)-set. However, since \(X_1\) per se is not a parametric unique \(\zeta\)-set, it cannot be said to be a parametric unique \((\zeta,\gamma)\)-set. On the other hand for a star \(S_{1,n}\), \(n\geq 3\) the set \(X_1=\{v_0\}\) is both a parametric \((\gamma,\zeta)\)-set and a parametric unique \((\zeta,\gamma)\)-set. Studying such parametric combinations for say parameters \(\rho_1(G)\) and \(\rho_2(G)\) requires that, \(\rho_1(G)=\rho_2(G)\).

Conjecture 1. If graph \(G\) has a pendent vertex then \(G\) has a unique \(\zeta\)-set if and only no \(\zeta\)-set exists which contains a pendent vertex.

A strict proof of Corollary 8 through mathematical induction is an interesting exercise for the reader.

Acknowledgments

The authors would like to thank the anonymous referees for their constructive comments, which helped to improve on the elegance of this paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

In 1939 Oldenburger considered, within the context of symbolic dynamics, a sequence having the property of coinciding with its own run length encoding [1]. If we choose the alphabet \(\left\lbrace 1,2 \right\rbrace\) there are two such sequences, the second of which being simply the first one without the initial element. The two sequences start as

\[1221121221\dots,\] and \[2211212212\dots\cdot\] In 1965 Kolakoski rediscovered the sequence [2], and it was easily established that it is not eventually periodic. Besides this, very little is known about the sequence. In particular, it is still not known whether it is recurrent, whether it has basic symmetry properties (mirror/reversal invariance) and whether the asymptotic density of 1s exists and equals \(\frac{1}{2}\), a conjecture formulated by Keane [3]. A sharp bound (\(0.5 \pm 0.00084\)) for the density of 1s has been provided [4]. Concerning other properties of the sequence, it has been proved that it is cube-free, which is a particular case of a more general result on repetitions [5]. Moreover, a measure conjectured to completely describe the densities of all subwords of the sequence has been introduced, and the conjecture has been proved under fairly natural additional hypotheses [6]. Recursive formulas for the \(n\)-th element of the sequence are also known [7,8].

The sequence is relevant for applications concerning optical properties of aperiodic structures [9,10,11], but probably its most interesting features are linked to the unique combination of the simplicity of its definition and the difficulty of the problems it raises.

The sequence is nowadays indexed as A000002 in Sloane's online Encyclopedia of Integer Sequences.

In this paper we study the open problems - namely recurrence, mirror and reversal invariance and asymptotic frequency of digits - trying to identify a unifying concept, i.e. the parity of the integrals (the converse transform of run length encoding) of subwords of the sequence. After introducing some notation and terminology in Section 2, the main open problems are reformulated in terms of parity of integrals of prefixes in Section 3, while in Section 4 we prove some results, including that recurrence implies reversal invariance, and provide sufficient conditions implying that the asymptotic frequency of 1s is \(\frac{1}{2}\). In Section 5 we describe a constructive procedure showing the existence of arbitrarily long recurrent subwords and identify places where they must occur in the structure of \(S\). Finally, in Section 6 we formulate some conjectures arising from the proposed approach.

We tried to make the paper as self-contained as possible. For this reason we included the proof of some facts (covered in Lemmas 1, 3, 8, 9 ,10 and 11) already known, although generally not presented, in the cited literature, exactly in the form proposed herein. In this way we could also ``optimize" their exact formulation for our aims.

2. Notation and preliminary definitions

Let \(\mathcal{A}^*\) be the set of finite words on the alphabet \[\mathcal{A}=\left\lbrace 1,2 \right\rbrace,\] and \(\mathcal{A}^{\infty}\) the set \(\mathcal{A}^*\cup \mathcal{A}^{\omega}\) of all finite or infinite words on \(\mathcal{A}\). We let \(\epsilon\) denote the empty word and we set \(\mathcal{A}^+:=\mathcal{A}^*\setminus \left\lbrace\epsilon\right\rbrace\).

The concatenation of the finite word \(w=a_1a_2\dots a_n\) and the (possibly infinite) word \(v=b_1b_2\dots\), i.e., the word \(a_1a_2\dots a_nb_1b_2\dots\), is written as \(wv\). The sets \(\mathcal{A}^*\) and \(\mathcal{A}^+\) have respectively the structure of a free monoid and a free semigroup with the internal operation defined as the concatenation of words.

For every \(w\in\mathcal{A}^{+}\), by \(\widetilde{w}\) we mean the mirror word of \(w\), i.e. the transform of \(w\) under the substitutions

\[1\to 2,\] \[2\to 1.\] We also set \(\widetilde{\epsilon}:=\epsilon\). If \(w=a_1\dots a_n\) is a word in \(\mathcal{A}^+\), we set \(\Sigma w:= \Sigma_{i=1}^n a_i\) and \(\overleftarrow{w}:=a_na_{n-1}\dots a_1\), calling the former sum of \(w\) and the latter inverse word of \(w\) (we set \(\overleftarrow{\epsilon}:=\epsilon\)). We indicate by \(|w|\) the length of \(w\), i.e., the positive integer \(n\) (we set \(|\epsilon|:=0\)).

We say that \(v\in\mathcal{A}^+\) is a subword of \(w=a_1a_2\dots\in \mathcal{A}^{\infty} \) if there exist a positive integer \(k\) and a non-negative integer \(h\) such that \(v=a_k a_{k+1}\dots a_{k+h}\). In the following it will be handy to have a term concisely referring to a particular occurrence of a subword. Therefore, we call the pair \((v,k)\) a subrow of \(w\) and say that \((v,k)\) is an occurrence of \(v\) in \(w\), and that two subrows \((v_1,k)\) and \((v_2,h)\) coincide as subwords if \(v_1=v_2\). When we want to emphasize the initial and final elements of the subrow \(a_ka_{k+1}\dots a_{k+h}\), we write it as \(w_{k,k+h}\). To lighten the notation, if there is no possibility of confusion, we may use the same symbol for the subrow \((v,k)\) and the subword \(v\). We let \(\mathcal{SR}(S)\) denote the set of all the finite subrows of \(S\).

We say that \(v\) is a prefix of a (possibly infinite) word \(w=a_1a_2\dots\) if there is a positive integer \(k\) such that \(v=a_1\dots a_k\). We say that \(v\) is a suffix of a finite word \(w=a_1\dots a_n\) if there is a positive integer \(k< n\) such that \(v=a_{n-k}a_{n-k+1}\dots a_n\).

Let us define a map from \(\mathcal{A}^{\infty}\) to itself by means of alternating substitution rules. Specifically, for every nonempty \(w\in \mathcal{A}^{\infty}\) we define the following substitution rules:

We let \(w^{-1}\) denote the transform of \(w\) under the substitutions \((1)^{1}\). We also set \(\epsilon^{-1}:=\epsilon\). For every \(w\in\mathcal{A}^{\infty}\), we define inductively: We refer to the \((\cdot)^{-1}\) map as the integration map.

Alternating substitution rules are quite well investigated and have also been generalized [12]. Relative results have been used specifically to study the Kolakoski sequence. It was indeed proved that, even if the rules (1) are very close to the simplest possible case of alternating substitution, its fixed point (i.e., Kolakoski sequence) cannot be obtained by iteration of a simple substitution [13].

The existence and uniqueness of the Kolakoski sequence are established by means of the following Lemma.

Lemma 1. There exists a unique element \(S\) of \(\left\lbrace 1,2 \right\rbrace^{\omega}\) such that \(S=S^{-1}\). Moreover, indicating the \(n\)-th element of \(S\) by \(s_n\), for every positive integer \(n\) there exists a positive integer \(h\) such that \(s_n\) is the \(n\)-th element of \((12)^{-k}\) for every integer \(k\) such that \(k\ge h\).

Proof. Take a finite word \(u\) such that \[u^{-1}=uv_1,\] with \(v_1\ne\epsilon\). Integrating both sides of the previous equality one gets \[u^{-2}=(uv_1)^{-1}=u^{-1}v_2=uv_1v_2,\] where \(v_2\) equals \(v_1^{-1}\) or \(\widetilde{v_1^{-1}}\) according to \(|u|\) being respectively even or odd. Iterating the argument it follows that, for every non-negative integer \(k\),

where, for \(2\le h\le k\), \(v_h=v_{h-1}^{-1}\) or \(v_h=\widetilde{v_{h-1}^{-1}}\) according to \(|uv_1\dots v_{h-2}|\) being respectively even or odd. Since the words \(v_i\) are nonempty, an arbitrarily long prefix of \(u^{-k}\) remains unaltered by further integrations. More precisely, if we write \(u^{-k}\) as \[u^{-k}=a_1^ka_2^k\dots a_{|u^{-k}|}^k,\] (where \(a_i ^k\in \left\lbrace 1,2\right\rbrace\)), for every positive integer \(n\) there is \(h\) such that \(a_n^{j_1}=a_n^{j_2}\) for every \(j_1,j_2\ge h\). Hence we can define the limit sequence \(S\) of the right hand side of (3) for \(k\to\infty\), which clearly verifies \(S=S^{-1}\). Taking \(u=12\) one gets the existence of \(S\). As for the uniqueness, it follows immediately observing that the only word of length 2 which is a prefix of its integral is 12.

Definition 2. The sequence \(S\) is called the Kolakoski sequence.

By the arbitrariness of the prefix \(u\) in the previous proof, we easily get by induction the following lemma:

Lemma 3. If \(p\) is a prefix of \(S\), such is \(p^{-k}\) for every non-negative integer \(k\).

We now want to adapt the previous definition of integral so that it applies nicely to subrows of \(S\), meaning that we can identify which subrow of \(S\) can be naturally seen as the integral of a given subrow. We thus want a version of the integration map which maps \(\mathcal{SR}(S)\) to itself (while \((\cdot)^{-1}\) maps \(\mathcal{A}^{\infty}\) to itself). Therefore we introduce the following definition:

Definition 4. Let \(w\) be a subrow of \(S\) and \(u\) the prefix such that \(S=uw\dots\). We define the \(S\)-integral of the subrow \(w\) as the subrow \(w^{-1}_{S}:=S_{h,k}\) where \[h=|u^{-1}|+1\quad\text{and} \quad k=|u^{-1}|+|w^{-1}|.\] We also define inductively \(w^{-n}_S:=\left(w^{-n+1}_S\right)^{-1}_S\).

Remark 1. Since \((\cdot)^{-1}\) is a non-morphic map, the \(S\)-integral of a subrow \(w\) does not coincide always with its integral as a subword, defined by means of the substitution rules (1). Indeed, if \(uw\) is a prefix of \(S\), considering \(w^{-1}_S\) as a subword, we have

\[w^{-1}_{S}=w^{-1},\] if \(|u|\) is even and \[w^{-1}_{S}=\widetilde{w^{-1}},\] if \(|u|\) is odd. Notice also that in general, for a subrow \(w\) which is not a prefix and for \(k\) large, \(w^{-k}\) and \(w_S^{-k}\) are different words which are not linked in any trivial way.

Next we want to define the property of a subrow of having \(S\)-integrals of even length up to a certain order, starting from the order 0 (that is, from the length of the subrow itself).

More precisely, we introduce the following definition:

Definition 5. We say that the subrow \(w\) is \(k\)-regular if \(|w^{-h}_S|\) is even for \(0\le h\le k\). We say that a subrow is \(k\)-normal if it is \(k\)-regular but not \((k+1)\)-regular. We say that a subrow is \(\infty\)-regular if it is \(k\)-regular for every non-negative integer \(k\).

Notice that in case \(w\) is a prefix, \(k\)-regularity reduces to requiring that \(|w^{-h}|\) is even for \(0\le h\le k\).

We indicate by \(k\)-R the subset of \(\mathcal{SR}(S)\) consisting of all the \(k\)-regular subrows of \(S\), by \(k\)-N the subset of \(\mathcal{SR}(S)\) consisting of all the \(k\)-normal subrows of \(S\) and by \(\infty\)-R the subset of \(\mathcal{SR}(S)\) consisting of all the \(\infty\)-regular subrows of \(S\). It is easily proved the following lemma:

Lemma 6.

• 1.   \(k\ge h \implies k\)-R \(\subseteq h\)-R.
• 2.   For every non-negative integer \(k\), \(k\)-N \(\subset k\)-R.
• 3.   \(k\ne h \implies k\)-N \(\cap\,\, h\)-N \(=\emptyset\).
• 4.   For every non-negative integer \(k\), \(w\in k\)-N \( \implies w \notin \infty\)-R.
• 5.   For positive integers \(a< b< c\), \(S_{a,b}, S_{(b+1),c} \in k\)-R \(\implies S_{a,c}\in k\)-R.
• 6.   For positive integers \(a< b< c\), \(S_{a,b}, S_{(b+1),c} \in k\)-N \(\implies S_{a,c}\in (k+1)\)-R.

Next we want to introduce the converse operation of integration, i.e., the so called derivative for words in \(\mathcal{A}^*\), which, roughly speaking, coincides with a run-length counting operation. However, we should take care to avoid the ambiguity arising when a subword starts or ends with a single digit not belonging to a pair of equal elements of the alphabet, as in that case we cannot know the length of its run without looking outside the subword. For this reason it is usual [6] to cut off those single digits, when they are present.

More precisely, for every \(w=a_1\dots a_n\in\mathcal{A}^+\) we define \(w'\) as the unique finite word such that \((w')^{-1}\) equals

\begin{align*} a_1\dots a_n\quad& \text{if}\quad a_1= a_2\quad \text{and} \quad a_{n-1}= a_n,\\ a_2\dots a_n \quad& \text{if}\quad a_1\ne a_2\quad \text{and}\quad a_{n-1}= a_n,\\ a_1\dots a_{n-1} \quad& \text{if}\quad a_1= a_2\quad \text{and}\quad a_{n-1}\ne a_n,\\ a_2\dots a_{n-1} \quad&\text{if}\quad a_1= a_2\quad \text{and}\quad a_{n-1}= a_n. \end{align*} We also set \(1'=2'=\epsilon':=\epsilon\), so that derivatives of every order exist for all elements of \(\mathcal{A}^*\); notice that this also implies \((12)'=(21)'=\epsilon\). We also define inductively \(w^{(n)}:=(w^{(n-1)})'\). Finally, we define the derivative of an infinite word \(v=a_1a_2\dots \in\mathcal{A}^{\omega}\), not ultimately constant, as the word whose \(m\)-th element is the \(m\)-th element of \((a_1\dots a_{n})'\) for all sufficiently large \(n\). It is easily seen that this \(m\)-th element stays the same when \(n\) diverges unless the sequence is ultimately constant (in which case the derivative of the sequence is not defined).

Adopting the usual convention, we indicate by \(C^k\) the set of words which belong to \(\mathcal{A}^{\infty}\) together with their first \(k\) derivatives, and by \(C^{\infty}\) the set \(\bigcap_{k\in\mathbb{N}} C^k\).

Similarly to what done before for integrals, we want now to adapt the definition of derivative so that it works for subrows. Therefore we introduce the following definition:

Definition 7. Let \(w\) be a subrow of \(S\). If there exists a subrow \(v\) such that \(v_S^{-1}=w\), we set \(w_S':=v\). We call \(v\) the \(S\)-derivative of \(w\).

We also define inductively \(w_S^{(n)}:=(w_S^{(n-1)})_S'\), of course if \(w_s^{(k)}\) admits an \(S\)-derivative for every \(k\le n\).

Remark 2. Notice that not every subrow has an \(S\)-derivative. For instance there is no subrow \(u\) such that \(s_{3}s_4=u_S^{-1}\).

The following Lemma characterizes the subrows admitting an \(S\)-derivative.

Lemma 8. Let \(n\) and \(m\) be two positive integers such that \(n< m\). If \(w=s_n\dots s_{m}\) is a subrow of \(S\), it admits an \(S\)-derivative if and only if \(s_{n-1}\ne s_n\) and \(s_{m}\ne s_{m+1}\). Moreover, if \(w_S'=s_h\dots s_{h+k}\) then \(h\le n\) and \(h+k\le m\).

Proof. Suppose that \(w\) admits an \(S\)-derivative. Then \(w\) is the transform under substitutions (1) (or the mirror of the transform under (1)) of another subrow \(v=s_h \dots s_j\) (\(j\ge h\)). This means in particular that \((s_j)_S^{-1}=s_m\) or \((s_j)_S^{-1}=s_{m-1}s_{m}\), so that \((s_{j+1})_S^{-1}=s_{m+1}\) or \((s_{j+1})_S^{-1}=s_{m+1}s_{m+2}\). Since \(j\) and \(j+1\) cannot be both even or both odd, it follows that \(s_m \ne s_{m+1}\). The proof proceeds analogously for \(s_n\) if \(s_{n-1}\) exists, otherwise (i.e., if \(w\) is a prefix), the thesis is vacuously true.

Conversely, suppose that \(s_{n-1}\ne s_n\) and \(s_{m}\ne s_{m+1}\). Then, by definition of \(S\), \(w\) is the transform under (1) (or the mirror of the transform under (1)) of some subrow \(v\).

Finally, if \(w_S'=s_h\dots s_k\), the inequalities \(h\le n\) and \(h+k\le m\) follow from the fact that, for every word \(w\), \(|w^{-1}|\ge |w|\).

Since \(w'\) is always a subword of \(w_S'\), the following Lemma is a consequence of the previous one:

Lemma 9. If \(w=s_ns_{n+1}\dots s_{n+m}\) is a subrow of \(S\) with nonempty derivative, there exists \(h\) and \(k\) such that \(w'=s_{h}s_{h+1}\dots s_{h+k}\).

The previous Lemma immediately implies that:

Lemma 10. If \(w\) is a subword of \(S\), then \(w\in C^{\infty}\).

Finally we define formally what is meant by asymptotic frequency of digits. Indicating by \(|v|_x\) the number of occurrences of the digit \(x\) (\(x\in\left\lbrace 1,2 \right\rbrace\)) in \(v\in \mathcal{A^+}\), and by \(f_v(x)\) the frequency of \(x\) in \(v\), i.e. the number \(\frac{|v|_x}{|v|}\), we set \[f_{\infty}(x):=\lim_{n\to \infty} \frac{|S_{1,n}|_{x}}{n},\] whether the limit exists. The most famous conjecture concerning Kolakoski sequence is Keane's conjecture [3]: \[f_{\infty}(1)\quad \text{exists and equals}\quad \frac{1}{2}.\]

3. Reformulation of the problems

In this section we reformulate some open questions concerning \(S\) in terms of regularity/normality of subrows. Let us recall that, according to (2), elements with even (odd) index in \(S\) are mapped by the integration in 2 or 22 (1 or 11). We use systematically this fact (usually without mentioning it explicitly) throughout.

In the following we will need a (rough) estimate of the relative length of \(w\), \(w'\) and \(w^{-1}\) when \(w\) is a subword of \(S\), ensuring in particular that, for every finite word \(w\) with more than one element,

and and that for every positive integer \(k\), This is obtained observing that \(|w^{-1}|=\sum w\), and that the maximum and minimum density of 2s in a \(C^{\infty}\) word are achieved respectively by \(11211\) and \(22122\). From this (recalling that the derivative cuts off single digits at both ends) the following Lemma is easily proved.

Lemma 11. If \(w\) is a subrow of \(S\) and \(|w|\ge 3\), then

\begin{align*} \frac{6}{5}|w| &\le |w^{-1}|=|w_S^{-1}|\le \frac{9}{5}|w|,\\ \frac{5}{9}|w| &\le |w_S'|\le \frac{5}{6}|w|\;\;\;(\text{if}\ \ w \ \ \text{admits}\,\, \text{an}\,\, S\text{-derivative}),\end{align*} and \[\frac{1}{4}|w|\le |w'|\le \frac{5}{6}|w|.\] The asymptotic behaviors (4), (5) and (6) immediately follow from Lemma 11 (notice that, if \(|w|=2\), \(|(w_S)^{-2}|\ge 3\)).

We add some definitions: a sequence \(w\in \mathcal{A}^{\omega}\) is called recurrent if every finite subword of it is repeated (and therefore every finite subword is repeated infinitely many times). It is called uniformly recurrent if it is recurrent and the gaps between consecutive occurrences of every given finite subword are bounded. Moreover, \(w\) is called mirror invariant (reversal invariant) if the set of its finite subwords is closed under the mirror operation: \(v\to \widetilde{v}\) (inverse operation: \(v\to \overleftarrow{v}\)).

It is a well known result that, for \(S\), mirror invariance implies recurrence [6]. The converse implication is not trivial, and sufficient conditions for it to hold are provided in Theorem 17. For this, though, we need some preliminary results.

The links between recurrence, mirror invariance and regularity/normality of subrows are established in the following Lemmas.

Lemma 12. \(S\) is recurrent if and only if for every positive integer \(k\) there is a \(k\)-regular prefix of \(S\).

Proof. Suppose that we can find a \(k\)-regular prefix \(w\) of \(S\) for every non-negative integer \(k\).

First of all notice that \(w\in k\)-R implies that \(|w|>2\), which in turn implies that \(|w^{-a}|\) is strictly larger than \(|w^{-b}|\) for every choice \(a,b\) of positive integers such that \(a>b\) (as there are no runs of consecutive 1s longer than 2 in \(w\)). From \(w\in k\)-R it follows that (as soon as \(k \ge 1\)) \(|w|\) and \(|w^{-1}|\) are even prefixes of \(S\), so that \(s_{|w|+1}\) and \(s_{|w^{-1}|+1}\) are both odd-indexed elements of \(S\). Then from Lemma 3, recalling the substitution rules (1), it follows that \(w^{-1}1\) and \(w^{-2}12\) are both prefixes for \(S\).

Integrating further we find a prefix of the form

where the last equality is again due to the fact that \(w\in k\)-R. By Lemma 3 the last factor in the right hand side of (7) coincides with a prefix of \(S\). Recalling (5), this prefix is arbitrarily long if \(k\) is large enough, which is sufficient to conclude that \(S\) is recurrent.

Conversely, suppose that \(S\) is recurrent. This implies, in particular, that arbitrarily long prefixes of \(S\) are repeated infinitely many times, thus for every positive integer \(N\) there is a prefix \(w\) and a prefix of the form \(wvw\) such that both \(|w|\) and \(|v|\) are larger than \(N\). By suitably selecting \(w\), we can assume that the last element of \(w\) is not equal to the first element of \(v\). Moreover, since \(w\) starts with 12211, by Lemma 10 the last element of \(v\) has to be different from the first element of \(w\), as otherwise \(vw\notin C^{\infty}\). Therefore, by Lemma 8, there exists a nonempty word \(u_1\) such that, setting \(p_1:=(w_S)'\), the word \(p_1 u_1 p_1\) is also a prefix for \(S\).

We then define recursively \[p_{i+1}:=(S_{1,|p_i|})_S'\] and \[u_{i+1}:=(u_i)_S'\] in case \(s_{|p_i|}\ne s_{|p_i|+1}\). If instead \(s_{|p_i|}= s_{|p_i|+1}\) we replace, in the definition of \(p_{i+1}\) and \(u_{i+1}\), \(p_i\) with the largest of its prefixes admitting an \(S\)-derivative and \(u_{i}\) with the smallest subrow having \(u_{i}\) as a suffix and admitting an \(S\)-derivative.

Since \(|v|\) can be arbitrarily large and recalling (6), we have that \[p_k u_k p_k\] is a prefix of \(S\) (with \(u_k\ne \epsilon\)) for every \(k\) such that \(|p_k|\ge 2\). Therefore \(p_k\) starts with 1 for every \(k\) such that \(|p_k|\ge 2\). Since \(p_k\) also follows the prefix \(p_ku_k\) in \(S\), this means, recalling the substitution rules (1), that the first \(k-1\) integrals of the prefix \(p_{k} u_{k}\) have even length. By Lemma 11 it follows that \(k \to \infty\) when \(|w|\to \infty\).

Lemma 13. \(S\) is mirror invariant if and only if for every positive integer \(n\) there is a \(k\)-normal prefix of \(S\) with \(k>n\).

Proof. Suppose that \(w\) is a \(k\)-normal prefix of \(S\). As seen in the proof of Lemma 12, \(w^{-h}(12)^{-h+2}\) is also a prefix for \(S\) for every \(h\le k+1\). Since \(|w^{-k-1}|\) is odd, integrating further and recalling Lemma 3 one gets the prefix \(w^{-k-2}\widetilde{v}\), where \(v=(12)^{-k}\). By Lemma 1, \(v\) is also a prefix of \(S\), and \(|v|\) is arbitrarily large if \(k\) is large enough. Since concatenation commutes with the mirror operation (that is: \(\widetilde{uv}=\widetilde{u}\widetilde{v}\) for every \(u,v \in \mathcal{A^+}\)), this is sufficient to conclude that \(S\) is mirror invariant.

Conversely, suppose that \(S\) is mirror invariant and let \(w\) be a prefix of \(S\) such that its last element is not equal to the following element of \(S\). By mirror invariance there is a prefix of the form \(wv\widetilde{w}\). Let us define the subwords \(p_n\) and \(u_n\) as done in the previous proof, and let \(\bar{n}\) be the largest integer for which \(|p_n|\ge 2\). Since \(S\)-derivatives of mirror words coincide as subwords, there are prefixes of the form \[p_n u_n p_n\] with \(u_n\) nonempty for every positive integer \(n\le\bar{n}\) (notice that this means that \(S\) is recurrent). As \(p_n\) is a prefix of \(w\) and so starts with 1 for every \(n\le\bar{n}\), (6) ensures that the prefix \(p_n u_n\) is \(k\)-regular for arbitrarily large positive integers \(k\) if \(|w|\) and \(|v|\) are chosen large enough. Moreover, since \(\widetilde{w}\) starts with 2 and \((p_1u_1p_1)^{-1}=wv\widetilde{w}\) by hypothesis, \(|p_1u_1|\) has to be odd, and therefore the prefix \(p_{\bar{n}}u_{\bar{n}}\) is \((\bar{n}-2)\)-normal, where \(\bar{n}-2\) can be arbitrarily large if \(|w|\) is chosen large enough.

It is known that mirror invariance is equivalent to: every \(C^{\infty}\) finite word occurs in \(S\) [6]. One implication is obvious, while the other (whose proof was left to the reader in [6]) easily follows from the fact that mirror invariance implies that, if \(w\) does not occur in \(S\), neither does \(w'\). This result and Lemma 13 mean that:

Lemma 14. Every \(C^{\infty}\) word is a subword of \(S\) if and only if for every positive integer \(n\) there is a \(k\)-normal prefix of \(S\) with \(k>n\).

Concerning uniform recurrence, we have the following lemma:

Lemma 15. \(S\) is uniformly recurrent if and only if, for every positive integer \(n\), \(S\) can be written as an infinite concatenation of \(k\)-regular subrows of bounded length with \(k>n\).

Proof. Suppose that, for every positive integer \(N\), there exists a sequence of subwords \(w_i\, (i\in \mathbb{N})\) and a positive integer \(M\) such that \(|w_i|< M\) for every \(i\) and

with every \(w_i\in k\)-R and \(k>N\). Then integrating (8) \(k\) times yields \begin{equation*} S=w_1^{-k}w_2^{-k}\dots. \end{equation*} Since \(|w_i^{-h}|\) is even for every \(h\le k\), the word \((12)^{-k+2}\) is a prefix of \(w_i^{-k}\) for every \(i\ge 2\), and by Lemma 1 it is also a prefix of \(S\). Since, by Lemma 11, \(|w_i^{-k}|< M\left(\frac{9}{5}\right)^k\), it follows that \(S\) is uniformly recurrent.

Conversely, suppose that \(S\) is uniformly recurrent. Then, for every prefix \(w\), \(S\) can be written as

with \(2< |u^i|< M\) for every \(i\) for some \(M>0\). We can assume that \(w\) ends with \(11\) or \(22\), so that its last element is not equal to the first element of \(u^i\) for every \(i\). Since \(w\) starts with \(12211\), every \(u^i\) has to end with \(2\), otherwise a subword which is not \(C^{\infty}\) would occur in \(S\), which is not possible by Lemma 10. By Lemma 8 we then have \begin{equation*} S=w_S'(u_1)_S'w_S'(u_2)_S'w_S'\dots \,. \end{equation*} Taking \(w\) long enough and defining the subwords \(p_i\) (\(i=1,\dots,k\)) as in the proof of Lemmas 12 and 13 and the subrows \(u_{i}^j\) (\(j=1,2,\dots\)) accordingly, we can iterate \(k\) times the argument, so as to obtain \begin{equation*} S=p_ku_k^1p_ku_k^2 p_k\dots. \end{equation*} Since \(M\) can be chosen arbitrarily large (by simply neglecting a suitable number of occurrences of \(w\) in \(S\) if needed), the limit behaviour (6) means that \(|u_k^j|\) can be made nonempty for every positive integer \(j\) and for arbitrarily large \(k\). Therefore, for every positive integer \(n\) and for arbitrarily large \(k\) we can find prefixes of the form \[p_ku_k^1\dots p_ku_k^n,\] and noticing that \(p_i\) begins with 1 for every positive integer \(i< k\), it follows that these prefixes are \((k-1)\)-regular. Then, by Lemma 6, so is every subrow \(S_{a,a+b}\) where \(a=|p_{k}u_{k}^h|+1\) and \(b=|p_{k}u_{k}^{h+1}|\) (\( h=1,2,\dots, k-1\)). Recalling the definition of \(p_i\) and \(u_i^j\), and that differentiation cannot increase the length of subrows, the inequalities \(|p_{k}|\le |w|\) and \(|u_{k}^j|\le |w|+M\) follow, so that each subrow of type \(p_{k}u_{k}^j\) has length not larger than \(2|w|+M\), which concludes the proof.

Remark 3. Lemmas 12, 13, 14 and 15 can be straightforwardly adapted to generalized Kolakoski sequences defined over binary alphabets \(\left\lbrace m,n\right\rbrace\) other than \(\mathcal{A}\) if we define analogously the concepts of regularity and normality of subrows.

On generalized Kolakoski words we mention the works by Sing [10,14] and, in an interesting but slightly different direction compared to typical Kolakoski literature, by Shen [15].

4. Main results

Let us start by observing that the existence of an \(\infty\)-regular prefix of \(S\) would have strong consequences on its structure and properties, as \(S\) would then be recurrent and would have a rigidly fractal structure.

More precisely, we establish the following result:

Theorem 16. Suppose that \(S\) has an \(\infty\)-regular prefix \(w\) and let \(k\) be a positive integer large enough so that \(|(12)^{-k+2}|>|w|\). Then, for every positive integer \(n\), \(S\) has a prefix with the following structure:

In particular, \(S\) is recurrent.

Proof. Since the prefix \(w\) is \(\infty\)-regular, there exists a positive integer \(\bar{h}\) such that \[w^{-h}(12)^{-h+2}\] is also a prefix for every \(h \ge \bar{h}\). Therefore arbitrarily long prefixes of \(S\) are repeated, which is enough to have recurrence. In particular, if \(k\) is such that \(|(12)^{-k+2}|>|w|\), then, by Lemma 1, \[w^{-k}w\] is a prefix. Integrating further for \(k\) times, and recalling that \(w\in \infty\)-R, we get the prefix \[w^{-2k}w^{-k}w,\] and continuing the integrations for further \((n-2)k\) times we get (10). Notice that, since \(w^{-nk}\) is a prefix for every \(n\), it has to begin with \(w^{-hk}\) for every \(h< k\).

Remark 4. Theorem 16 can be applied to generalized Kolakoski words. Its application to Kolakoski words over binary alphabets \(\lbrace m,n \rbrace\) in which \(m\) and \(n\) are both even or both odd (and therefore every prefix of even length if \(\infty\)-regular) immediately implies that those sequences are recurrent, which is a well-known result already obtained by other means [14].

As already said, it is a known result that if \(S\) is mirror invariant, then it is recurrent [6] (notice that Lemmas 12 and 13 immediately imply that). The converse implication is obtained with an additional hypothesis in the following theorem:

Theorem 17. If \(S\) is recurrent and \(\infty\)-R \(\,=\emptyset\), then \(S\) is mirror invariant.

Proof. Suppose that \(\infty\)-R\(\,=\emptyset\) and that \(S\) is recurrent. Then by Lemma 12 there is a strictly increasing sequence of positive integers \(k_n\) such that \(S\) has a \(k_n\)-regular prefix \(w_n\) for every \(n\). Since \(w_n\notin \infty\)-R, there is a positive integer \(k\) which is the least integer such that \(|w_n^{-k}|\) is odd. As seen in the proof of Lemma 12, \(w_n^{-k}(12)^{-k+2}\) is also a prefix of \(S\), and integrating once more (recalling Lemma 3) we get \[S=w_n^{-k-1}\widetilde{(12)^{-k+1}}\dots.\] Recalling that \((12)^{-k+1}\) is also a prefix of \(S\) by Lemma 1, and that \(k\) can be taken arbitrarily large by suitably choosing \(w_n\), we can conclude.

The implication from reversal invariance to recurrence is a known result holding for all elements of \(\mathcal{A}^{\omega}\) (for the application to differentiable sequences see [16], where a stronger result is proved, namely that recurrence of \(S\) is implied by the existence of arbitrarily long palindromes). The converse implication is proved in the following theorem:

Theorem 18. \(S\) is recurrent \(\implies\) \(S\) is reversal invariant.

Proof. Suppose that \(S\) is recurrent. Then, by Lemma 12, for every integer \(k\) the sequence \(S\) has a \(k\)-regular prefix \(w_k\). We have

where it is easily seen that \(w_k^{-2}\) must end with 2. Defining \(v\) by \(v2=w_k^{-2}\) we can write Integrating (12), and recalling that \(v2\in (k-2)\)-R, we have and as \(|v2|\) and \(|v212|\) are both even, the 2 appearing as the last element of \(v2\) is transformed by the rules (1) in the same way as the 2 appearing as the last element of \(v212\). Therefore prefix \((v2)^{-1}1\) can be rewritten as \begin{equation*} v^{-1}\overleftarrow{(12)^{-1}}. \end{equation*} Proceeding by induction, suppose that the prefix \((v2)^{-h}1\) can be rewritten as Integrating \(h\) times (12), and recalling that \(v2\) is \((k-2)\)-regular, we get the prefix Set \(n:=|(v2)^{-h}|\). Comparing the prefixes (14) and (15), it follows that, for every integer \(j\) such that \(0\le j\le |(12)^{-h}|\), the \((n-j)\)th element of \((v2)^{-h}\) is equal to and has always the same parity as the \((j+2)\)th element of \((12)^{-h}\), and therefore is transformed by the rules (1) in the same way. Therefore, since integrating (15) we get \((v2)^{-h-1}(12)^{-h-1}\), integrating (14) we must get the prefix Since \((12)^{-k}\) is a prefix of \(S\) for every \(k\) by Lemma 1, the arbitrariness of \(k\), and thus of \(h\), allows us to conclude that \(S\) is reversal invariant.

A natural question is whether, for a subrow \((w,n)\), the property of being \(k\)-regular for large values of \(k\) is compatible with the requirement \(w\in C^\infty\). In fact it is possible to prove more, i.e., that \(S\) can be eventually written as a concatenation of arbitrarily regular subrows. More precisely, we have the following theorem:

Theorem 19. For every non-negative integer \(n\), there exist a finite word \(u_n\) and finite words \(w_i\) (\(i=1,2\dots\)) such that

where the subrows \(w_i\in k\)-R for every \(i\) and \(k\ge n\).

Proof. We proceed by induction. Let us suppose that \(S\) has the form (17) and that \(w_i\in k\)-R \(\forall i\). Let \(M\) be the set of positive integers \(i_m\) such that \(w_{i_m}\notin (k+1)\)-R. Clearly the subrows \(w_{i_m}\) are \(k\)-normal. If \(M\) is finite, we define \(p:=\max\left\lbrace j\in\mathbb{N}^+:j \in M\right\rbrace\) and \(u_{n+1}:=u_nw_1\dots w_p\) so that \[S=u_{n+1}w_{p+1}w_{p+2}\dots\quad (p=1,2\dots),\] which is the desired result.

If \(M\) is infinite, \(i_m\) is a subsequence of \(i\), so for every positive integer \(m\) we can define the words \(v_m:=w_{i_m}w_{(i_m +1)}\dots w_{i_{(m+1)}}\). Every \(v_m\) is a concatenation of the \(k\)-normal subrow \(w_{i_m}\), the (possibly empty) word formed by the \(i_{(m+1)}-i_m-1\) words \(w_{i_m+h}\) (\(h=i_m+1\dots i_{m+1}-1\)), which are \((k+1)\)-regular, and the \(k\)-normal subrow \(w_{i_{(m+1)}}\). Therefore, by Lemma 6, \(v_m\in(k+1)\)-R for every \(m\), and therefore defining \(p:=\min M\) and the word \(u_{n+1}:=u_nw_1w_2\dots w_{p-1}\), we have

which is the desired result.

Finally, \(S\) is obviously written as a concatenation of 0-regular subrows, so the proof is concluded.

Remark 5. In the previous Theorem, if we start the inductive construction of the words \(w_i\) from \(u_0:=\epsilon\) and \(w_i:=s_{2i-1}s_{2i}\) (\(i=1,2\dots\)), we can have two different possibilities:

• 1.   \(u_n=\epsilon\) for every non-negative integer \(n\). In this case \(S\) is written as a concatenation of \(k\)-regular subrows for every \(k\), and therefore it is recurrent and reversal invariant by Lemmas 12 and 13.
• 2.   At some step \(\bar{n}\) of the inductive procedure we have \(\epsilon\ne u_{\bar{n}}\in (\bar{n}-1)\)-N. By Lemma 6 it follows that in this case, continuing the inductive procedure, we have \(u_n\in (\bar{n}-1)\)-N for every \(n>\bar{n}\).

Remark 6. We can apply the iterative procedure of Theorem 19 starting from an arbitrary element of \(S\), as no special properties of the beginning of \(S\) were used. This means that, for every positive integer \(k\), the same conclusion of the Lemma applies to the sequence \(s_k s_{k+1}s_{k+2}\dots\) .

To proceed further we need one more definition, as we want to assign a special name to the subrows \(v_i\) in (3). We recall that, for every prefix \(w\) of \(S\) we have, by Lemma 3, that \(w^{-k}\) is also a prefix.

Definition 20. For every prefix \(w\) such that \(|w|>1\), and for every positive integer \(k\), we define the \(k\)-th block generated by the prefix \(w\) as the unique subrow \(b_k\) such that \(w^{-k+1}b_k=w^{-k}\). We have clearly

and Notice that, for every \(k\), \(b_{k+1}=\left(b_k\right)^{-1}_S\) so that, for every \(k\): \[b_k=(b_{k-1})^{-1} \quad \text{if}\quad |wb_1\dots b_{k-2}|\quad\text{is even},\] \[b_k=\widetilde{(b_{k-1})^{-1}} \quad \text{if}\quad |wb_1\dots b_{k-2}|\quad\text{is odd}.\] Since \(|w|=|\widetilde{w}|\) for every finite word \(w\), we have by Lemma 11, that and therefore A block has the property of being always ''not too small" with respect to whatever comes before it in the sequence, and therefore the asymptotic frequencies of 1s and 2s, if they exist, have to be reached uniformly on the blocks. More precisely, we have the following lemma:

Lemma 21. Let \(w\) be a prefix (\(|w|\ge 2\)) of \(S\) and \(b_k\) (\(k=1,2\dots\)) the blocks generated by \(w\). Suppose \(f_{\infty}(1)\) exists. Then for every \(\epsilon>0\) there is an integer \(n\) such that \(|f_{b_k}(1)-f_{\infty}(1)|< \epsilon\) for every \(k\ge n\).

Proof. For every positive integer \(k\), let us define the prefixes \(u_k:=wb_1\dots b_k\). Using (22), it can be shown that there exist two real numbers \(c_1\) and \(c_2\), with \(0< c_1< c_2< 1\) such that, for every \(k\) large enough,

Indeed, setting \(\rho_k:= \frac{|b_{k+1}|}{|b_{k}|}\), we have \begin{equation*} \frac{|u_k|}{|b_k|}=\frac{|w|+|b_1|\left(1+\sum_{j=1}^{k-1}\prod_{i=1}^{j}\rho_i\right)}{|b_1|\prod_{i=1}^{k-1}\rho_i}\,, \end{equation*} so that, noticing that by Lemma 11 we have \(|w^{-1}|=|w|+|b_1|\le 6|b_1|\), it follows \begin{equation*} \frac{|u_k|}{|b_k|}\le 1+ \frac{6}{\prod_{i=1}^{k-1}\rho_i}+\sum_{j=2}^{k-1}\left(\prod_{i=j}^{k-1}\rho_i\right)^{-1}\,. \end{equation*} Since \(\rho_i\ge \frac{6}{5}\) for every positive integer \(i\), it follows \begin{equation*} \frac{|u_k|}{|b_k|}\le 1+6\left( \frac{5}{6} \right)^{k-1}+\sum_{i=1}^{k-2}\left(\frac{5}{6} \right)^{i} \xrightarrow[k \to \infty]{} 6\,. \end{equation*} Therefore, the optimal \(c_1\) is bounded from below by \(\frac{1}{6}\) and in particular is bounded away from zero (it is proved similarly that the optimal \(c_2\) is bounded away from 1). We have Since \(u_k\) is a prefix for every \(k\) and by (5) \(|u_k|\to \infty\) when \(k\to\infty\), if there exists \(f_{\infty}(1)\) then for every \(\epsilon>0\) there is \(h\) so large that, for every \(k>h\), \[f_{u_{k-1}}(1)=f_{\infty}(1)+{\epsilon}_1,\] and \[f_{u_{k}}(1)=f_{\infty}(1)+{\epsilon}_2,\] with \(\max\left\lbrace\epsilon_1,\epsilon_2\right\rbrace< \epsilon\). Therefore, from (24) we have \begin{equation*} f_{b_k}(1)c_1\le f_{u_k}(1)-f_{u_{k-1}}(1) (1-c_2)=\epsilon_2-\epsilon_1+f_{\infty}(1)c_2+\epsilon_1 c_2, \end{equation*} hence \begin{equation*} f_{b_k}(1)c_1-f_{\infty}(1)c_2\le \epsilon_2+\epsilon_1(1-c_2), \end{equation*} so that If there exists \(f_{\infty}(1)\) the difference \(c_2-c_1\) becomes arbitrarily small when \(k\) diverges. Indeed The right hand side of (26) tends to \(2-f_{\infty}(1)=1+f_{\infty}(2)\) when \(k\to \infty\). Therefore also \(\frac{|b_k|}{|u_k|}\) converges to a limit when \(k\) diverges, as it is obviously \(|u_k|=|u_{k-1}|+|b_k|\). Therefore, if \(k\) is large enough, we can take \(c_1\) and \(c_2\) such that \(c_2-c_1\) is arbitrarily small, and since \(c_1\) is bounded away from 0, (25) implies that \(|f_{b_k}(1)-f_{\infty}(1)|\) is vanishingly small when \(k\) diverges.

Remark 7. Let us take another prefix \(v\) with \(|v|>|w|\) and let \(d_k\) denote the blocks generated by \(v\). Then instead of (26) we have

where \(p_k:=vd_1\dots d_k\). Clearly the right hand side of (27) converges to \(1+f_{\infty}(2)\) faster than the right hand side of (26), as for every \(k\) we have \(|p_k|>|u_k|\). From this it easily follows that for every \(\epsilon>0\), if \(n\) satisfies Lemma 21 for a given prefix \(w\), then it satisfies Lemma 21 for \(v\), i.e., \(|f_{b_k}(1)-f_{\infty}(1)|< \epsilon\) for every \(k\ge n\) implies \(|f_{d_k}(1)-f_{\infty}(1)|< \epsilon\) for every \(k\ge n\).

Definition 22. We say that a prefix is \(k\)-minimal if it is the shortest \(k\)-normal prefix of \(S\).

Clearly for every positive integer \(k\) there is at most one \(k\)-minimal prefix.

We want now to provide sufficient conditions (as weak as possible) implying that Keane's conjecture is true. Specifically, we require the existence of arbitrarily normal prefixes as well as a "relaxed uniformity" property, i.e., that a sufficiently large portion of every block belonging to a certain subset (the ones generated by \(k\)-minimal prefixes) is representative of the frequency of 1s on that block.

More precisely, we have the following theorem:

Theorem 23. Suppose that

• 1.   There exists \(f_{\infty}(1)\);
• 2.   There is a strictly increasing sequence of positive integers \[K:=k_1,k_2,k_3\dots\] such that there exists a \(k_n\)-normal prefix \(p_{k_n}\) of \(S\) for every \(n\);
• 3.   For every \(\epsilon>0\), there is a positive integer \(L_\epsilon\) such that \(|f_{b_{m}^n}(1)-f_{c_{m}^n}(1)|< \epsilon\) for every positive integer \(m\) such that \(|b_m^n|>L_\epsilon\), where
  • \(b_{m}^n\) (\(m=1,2,\dots\)) are the blocks generated by the \(k_n\)-minimal prefixes \(p_{k_n}\),
  • \(c_{m}^n\) is a prefix of the block \(b_{m}^n\) such that \(|c_{m}^n|\ge L_\epsilon\).

Then \(f_{\infty}(1)=\frac{1}{2}\).

Proof. Take \(\epsilon>0\) and consider a prefix \(w\) so long that \(|w|\ge L_\epsilon\) and

for every prefix \(v\) such that \(|v|\ge|w|\). We can find a \(k_n\)-minimal prefix \(p\) with \(k_n\) being the smallest element of the sequence \(K\) such that \(w\) is a prefix of \((12)^{-k_n+2}\), which implies that \(p^{-k_n}w\) is a prefix of \(S\). By Lemma 21 there is a positive integer \(m\) such that the \(m\)-th element \(k_m\) of the sequence \(K\) has the property that \(|f_{b_j}(1)-f_{\infty}(1)|< \epsilon\) for every \(j\ge k_m\), where \(b_j\) are the blocks generated by \(p\). There are two possibilities:

• 1.   \(k_m>k_n\).

  Then by hypothesis 2. we can find a \(k_s\)-minimal prefix \(q\) where \(k_s\) is the smallest element of the sequence \(K\) such that \(k_s\ge k_m\) and \(|q|>|p|\) (of course the latter is verified for some \(k_s\) because there are only finitely many prefixes which are shorter than \(p\)). Clearly \(q^{-k_s}w\) is also a prefix of \(S\). Moreover, recalling Remark 7 and denoting by \(d_j\) the blocks generated by \(q\), we have

  \begin{equation} |f_{d_j}(1)-f_{\infty}(1)|< \epsilon, \label{d_k} \end{equation}

  (29)
  for every \(j\ge k_m\) and thus in particular for \(j\ge k_s\). We can assume that \(k_s\ge 2\), so that \(|q|\ge 16\), \(|d_1|\ge 8\) and thus
  \begin{equation} q^{-k_s}d_{k_s+1}=q^{-k_s}wu, \label{eqx} \end{equation}

  (30)
  with \(u\) nonempty.

  Since \(|q^{-k_s-1}|\) is odd by hypothesis, integrating two more times (30) we get the prefix

  \[q^{-k_s-2}d_{k_s+3}=q^{-k_s-2}\widetilde{w^{-2}}{u_S^{-2}},\] where \(u_S^{-2}\) is nonempty, so that \(\widetilde{w^{-2}}\) is a prefix of \(d_{k_s+3}\) (and does not coincide with it). Since \(|w^{-2}|>|w|\ge L_\epsilon\), by assumption the following inequalities are verified:
  \begin{equation} |f_{d_{k_s+3}}(1)-f_{\infty}(1)|< \epsilon, \label{comparings} \end{equation}

  (31)
  \begin{equation} |f_{w^{-2}}(1)-f_{\infty}(1)|< \epsilon, \label{comparing} \end{equation}

  (32)
  and moreover, by hypothesis 3, \[|f_{\widetilde{w^{-2}}}(1)-f_{d_{k_n+3}}(1)|< \epsilon.\] Combining the last inequality with (31) and (32), we get that the difference \(|f_{\widetilde{w^{-2}}}(1)-f_{w^{-2}}(1)|\) is vanishingly small if \(n\) is large enough, and therefore so is, by definition of mirror word, the difference \(|f_{w^{-2}}(1)-f_{w^{-2}}(2)|\), from which we can conclude.
• 2.   \(k_n\ge k_m\).

  Then we proceed as above with the prefix \(p\) instead of \(q\) and \(k_n\) instead of \(k_m\).

5. An iterative procedure providing arbitrarily long recurrent subwords

In this section we want to use the iterative construction shown in the proof of Theorem 19 to establish constructively the existence of recurrent subwords of arbitrary length and identify places where they must appear in the structure of \(S\). Before this, let us expicitly recall that, for any aperiodic sequence and every positive integer \(n\), the existence of at least \(n+1\) distinct subwords of length \(n\) (which are easily shown to be recurrent) is a basic combinatorial result (see for instance [17]). However, being so general, this kind of argument is of course non-constructive, not providing any insight about where to find such recurrent subwords in the sequence, nor about the structure of such subwords themselves.

In order to obtain a bit more, let us start by associating to every subrow of \(S\) a sequence over \(\left\lbrace 0,1\right\rbrace^{\omega}\) describing the parity of all its \(S\)-integrals. More precisely, we introduce the following definition:

Definition 24. For every subrow \(w\) we define \(P_0(w)=0\) if \(|w|\) is even, \(P_0(w)=1\) otherwise. We define inductively \(P_n(w)=0\) if \(|w^{-n+1}_S|\) is even, \(P_n(w)=1\) otherwise. We call the sequence \(P_n(w)\) the history of parity of the integrals of \(w\).

In the particular case in which \(w\) is a prefix, \(P_n(w)\) is simply the sequence describing the parities of \(|w^{-n}|\).

Lemma 25. Suppose that \(u_1\) and \(u_2\) are distinct prefixes of \(S\) such that \(S=u_1w_1\dots\) and \(S=u_2w_2\dots\) with the subrows \(w_1\) and \(w_2\) coinciding as subwords. If there exists \(N\) such that \(P_{N}(u_1)\ne P_{N}(u_2)\), then \((w_1)_S^{-k}\ne (w_2)_S^{-k}\) for every \(k\ge N\).

Proof. Suppose in particular that \(N\) is the least integer for which \(P_n(u_1)\ne P_n(u_2)\). Then it follows that \((w_1)^{-N+1}_S=(w_2)^{-N+1}\), while, recalling the substitution rules (1), we have \((w_1)^{-N}_S= \widetilde{\left(w_2\right)^{-N}_S}\ne (w_2)^{-N}_S\). To conclude it is enough to observe that if \(w\) and \(v\) are nonempty subwords, \(w\ne v\) implies that the four words \(w^{-1}\), \(\widetilde{w^{-1}}\), \(v^{-1}\) and \(\widetilde{v^{-1}}\) are all distinct.

Let us now start the inductive procedure described in the proof of Theorem 19 with the empty prefix \(u_0=\epsilon\) and \(w_i:=s_{2i-1}s_{2i}\) (\(i=1,2\dots\)). Suppose that \(\bar{n}\) is the least integer for which \(u_{\bar{n}}\ne \epsilon\) (we recall that, by Lemma 12, if \(u_n=\epsilon\) for every positive integer \(n\), then every subword of \(S\) is recurrent). It follows from the construction of Theorem 19 that it has to be \(u_{\bar{n}+k}\in\bar{n}\text{-}N\) for every positive integer \(k\). By direct inspection it can be seen that \(\bar{n}\) is not smaller than 2, as the prefix \(s_1s_2\dots s_{16} = 12 21 12 12 21 22 11 21\) is 2-regular. Therefore, according to Theorem 19, we have that, for every positive integer \(k\),

with \(u_k\in h\)-N (\(h\ge 2\)) and \(w_i\in k\)-R for every \(i\). Since \(u_k\) is 1-regular, we have that, writing \(S\) as the subrows \((w_i)_S^{-2}\) all begin with 12. Therefore integrating \(k-2\) times (34) and suitably defining the subrows \(v_i\), we obtain for \(S\) the structure \begin{equation*} S=v_0(z_1)_S^{-k+2}v_1(z_2)_S^{-k+2}\dots \end{equation*} were \(z_i=12\) for every integer \(i\), and the subrows \((z_i)_S^{-k+2}\) are all coinciding as subwords since, for every \(j\le k\), the parity of \(|(u_kw_1w_2\dots w_n)^{-j}|\) is the same for every \(n>0\). Finally, since \(k\) is arbitrarily large by Theorem 19, (5) implies that the recurrent subwords \((z_i)_S^{-k+2}\) are arbitrarily long.

We can also use the same iterative procedure to identify other arbitrarily long recurrent subwords which, in general, are not coinciding with the previous ones. Indeed, recalling Remark 6, we can also apply the iterative construction starting right after any given prefix of \(S\). Taking, for instance, the 1-normal prefix \(p:=\)1221, for every positive integer \(k\) we can write

\begin{equation*} S=p\bar{u}_k \bar{w}_1 \bar{w}_2\dots, \end{equation*} where, noticing that \(s_5\dots s_8\in 2\)-R, we have \(\bar{u}_k\in \bar{h}\)-N (\(\bar{h}\ge 2\)) and \(\bar{w}_i\in k\)-R for every \(i\). Since \(p\bar{u}_k\) is 1-regular by Lemma 6, we have that, writing \(S\) as the subrows \((\bar{w}_i)_S^{-2}\) all begin with 12. Therefore, integrating (35) \(k-2\) times and suitably defining the subrows \(\bar{v}_i\), we obtain for \(S\) the structure \begin{equation*} S=\bar{v}_0(\bar{z}_1)_S^{-k+2}\bar{v}_1 (\bar{z}_2)_S^{-k+2}\dots \end{equation*} were \(\bar{z}_i=12\) for every integer \(i\), and the subrows \((\bar{z}_i)_S^{-k+2}\) are all coinciding as subwords since, for every \(j\le k\), the parity of \(|(p\bar{u}_k\bar{w}_1\bar{w}_2\dots \bar{w}_n)^{-j}|\) is the same for every \(n>0\). Since by construction \(P_n(u_k)\ne P_n(p\bar{u}_k)\), Lemma 25 ensures that \((z_i)_S^{-k+2}\ne (\bar{z}_i)_S^{-k+2}\), while again since \(k\) can be arbitrarily large (from Theorem 19), the recurrent subwords \((\bar{z}_i)_S^{-k+2}\) have arbitrarily large length by (5).

6. More open questions

How do \(k\)-regular prefixes (or, in general, subrows) look? This is a difficult question. Let us take a look at the first cases. A prefix \(w=s_1\dots s_n\) is

• 0-regular if \(n\) is even;
• 1-regular if it is 0-regular and \(\Sigma w\) is even;
• 2-regular if it is 1-regular and \(\Sigma_{i=0}^{\frac{n}{2}-1} s_{2i+1}\) is even;
• 3-regular if it is 2-regular and \begin{align*} &\Big|\left\lbrace s_j: s_j=2\,\text{and}\,\, j\,\, \text{is odd} \right\rbrace\Big|+\Big|\left\lbrace s_j : s_j=1,\,\, j\,\, \text{is odd} \,\,\text{and}\,\, \Sigma_{i=1}^{j-1} s_i \,\,\text{is even}\right\rbrace\Big| \end{align*} is even.

With some effort one can go a bit further, but it is not easy to see where the thing is going.

From numerical computations one gets the impression that the requirement of being \(k\)-regular for large \(k\) is quite hard to meet - and we recall that recurrence for \(S\) is equivalent to the existence of arbitrarily regular prefixes. For instance, the shortest 10-regular prefix has length 6410, while the 10-minimal prefix has length 7144. Since the number of independent conditions that a finite word has to satisfy to be \(k\)-regular seems to increase with \(k\), the following conjecture arises naturally.

Conjecture 1. There are no \(\infty-\)regular subrows in \(\mathcal{SR}(S)\).

A consequence of this conjecture is seen in Theorem (17). It is also natural, in our view, to formulate a stronger conjecture, namely that two subrows whose integrals have exactly the same history of parity, must coincide. More precisely, we state the following conjecture:

Conjecture 2. If \(u\) and \(w\) are two subrows and \(P_n(u)=P_n(w)\) for every \(n\ge 0\), then \(u=w\).

If this is true, then Conjecture 1 follows, as if \(w\) is an \(\infty\)-regular subrow, we can split it in two subrows with the same history of parity, which contradicts Conjecture 2.

Acknowledgments

The author is deeply grateful to Lucio Russo (who introduced me to the problem), Stefano Isola and Riccardo Piergallini for many fruitful discussions.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

The concept of graph energy was introduced in 1978 [1], as a kind of generalization of the HMO total \(\pi\)-electron energy [2]. After a twenty-years period of silence, graph energy became a popular topic of research in chemical graph theory [3], with over than hundred papers published each year [4, 5 ]. Of the numerous properties established for graph energy, one remains a long-time open problem. Namely, already in the early days of the study of graph energy and its predecessor total \(\pi\)-electron energy, it was observed that it somehow decreases with the increasing number of zero eigenvalues in the graph spectrum [2, 6].

Let \(G\) be a (molecular) graph, possessing \(n\) vertices. Let \(\lambda_1,\lambda_2\ldots,\lambda_n\) be the eigenvalues of the \((0,1)\)-adjacency matrix of \(G\), forming the spectrum of \(G\). Then the energy of \(G\) is defined as [1, 3]: \[ E = E(G) = \sum_{i=1}^n |\lambda_i|, \] whereas its nullity, denoted by \(\eta=\eta(G)\), is equal to the number of zero eigenvalues (= the algebraic multiplicity of the number zero in the graph spectrum). Then the above mentioned regularity can be formally stated as follows.

Conjecture 1. Let \(G\) and \(G^\prime\) be two structurally similar graphs. Let \(\eta(G) < \eta(G^\prime)\). Then \(E(G) > E(G^\prime)\).

The problem with this conjecture is that the meaning of ``structurally similar graphs'' is not clear and cannot be rigorously defined. Earlier attempts to justify its validity were based on designing approximate expression for \(E(G)\), containing the term \(\eta(G)\) [6, 7, 8, 9], or by constructing a suitably chosen example for the graphs \(G,G^\prime\) [10].

It this paper we elaborate a general method for quantifying the dependence of graph energy on nullity, in which the usage of the ``structurally similar graph'' \(G^\prime\) is avoided.

2. Effect of nullity on the energy of a non-singular graph

Let \(G\) be a (molecular) graph as defined above, and assume that it is non-singular, i.e., that it has no zero eigenvalues, \(\eta(G)=0\). Let its characteristic polynomial be \[ \phi(G,\lambda) = \sum_{k=0}^n c_k\,\lambda^{n-k} = \prod_{i=1}^n (\lambda-\lambda_i), \] and recall that and for \(k=2,...,n\). In particular, \[ c_n = \prod_{i=1}^n \lambda_i \neq 0. \] Throughout this paper, since we are mainly interested in molecular graphs, it will be assumed that \(n\) is even, and that \[ \lambda_1 \geq \lambda_2 \geq \cdots \lambda_{n/2} > 0 > \lambda_{n/2+1} \geq \lambda_{n/2+2} \geq \cdots \geq \lambda_n\,. \] If so, then \(c_2<0\), \(c_4>0\), \(c_6<0\), \(c_8>0\), \ldots, i.e., and The coefficients \(c_k\) are the structural parameters of the graph \(G\) on which its energy depends. It is known that [11, 12, 13, 14]: where \(i=\sqrt{-1}\) and \(Re\) indicates the real part of a complex number. Recall that for any complex number \(z\), \ \(|z| \geq Re(z)\). Our approach is based on the following. Instead of looking for a graph \(G^\prime\) that would be ``structurally similar'' to \(G\), we construct a quasi-spectrum, consisting of the numbers \(\lambda^\prime_1,\lambda^\prime_2,\ldots,\lambda^\prime_n\), that would be as similar as possible to the true spectrum of \(G\), but that would contain a single zero element. Let \(\lambda^\prime_k \neq 0\) for \(k=1,2,\ldots,n-1\) and \(\lambda^\prime_n=0\), which is equivalent to the condition \(\eta(G^\prime) = 1\). Then, in analogy to Eqs. (1) and (2), we define \[ c^\prime_0 = 1 \ \ , \ \ c^\prime_1 = \sum_{i=1}^n \lambda^\prime_i = 0, \] and \[ c^\prime_k = \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} \lambda^\prime_{i_1}\,\lambda^\prime_{i_2}\cdots \lambda^\prime_{i_k}, \] for \(k=2,...,n\), implying that \[ c^\prime_n = \prod_{i=1}^n \lambda^\prime_i = 0\,. \] Bearing in mind Equation (5), we now construct an auxiliary quantity [15, 16]: \[ E^\prime(G) = \frac{2}{\pi} \int\limits_0^\infty \ln \left| \sum_{k=0}^n c^\prime_k\,(ix)^{-k} \right| dx\,. \] By requiring that \(c^\prime_k = c_k\) for \(k=1,2,\ldots,n-1\), it immediately follows that \[ E^\prime(G) = \frac{2}{\pi} \int\limits_0^\infty \ln \left| \sum_{k=0}^{n-1} c_k\,(ix)^{-k} \right| dx\,. \] Thus, the quantity \(E^\prime(G)\) can be calculated from the coefficients of the characteristic polynomial of the graph \(G\). The difference \(E(G)-E^\prime(G)\) can be understood as the effect of a single zero eigenvalue, i.e., nullity \(\eta=1\), on the energy of a non-singular graph \(G\). Using the Coulson--Jacobs formula [11, 12], we get \[ E(G)-E^\prime(G) = Re \left[ \frac{2}{\pi}\,\int\limits_0^\infty \ln \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}} { \sum\limits_{k=0}^{n-1} c_k\,(ix)^{n-k}}\,dx \right] = \frac{2}{\pi}\,\int\limits_0^\infty \ln \left| \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}} { \sum\limits_{k=0}^{n-1} c_k\,(ix)^{n-k}} \right| dx, \] i.e., From Equation (6) it follows that \(E(G)-E^\prime(G)>0\), in agreement with Conjecture 1. In order to see this, note that \[ \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}}{\sum\limits_{k=0}^{n-1} c_k\,(ix)^{n-k}} = \frac{A(x)-i\,B(x) + (-1)^{n/2}\,c_n}{A(x)-i\,B(x)}, \] where \( A(x) = \sum\limits_{k=0}^{n/2-1} (-1)^k\,c_{2k}\,x^{n-2k}\) and \(B(x) = \sum\limits_{k=0}^{n/2-1} (-1)^k\,c_{2k+1}\,x^{n-2k+1}\,. \) Then \[ Re \left[ \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}}{\sum\limits_{k=0}^{n-1} c_k\,(ix)^{n-k}} \right] = \frac{A(x)^2+B(x)^2 + (-1)^{n/2}\,c)n}{A(x)^2+B(x)^2} = 1 + \frac{(-1)^{n/2}\,c_n\,A(x)}{A(x)^2+B(x)^2}, \] which by relations (3) and (4) is greater than unity for all values of \(x \in [0,+\infty)\). Therefore, \begin{eqnarray*} E(G)-E^\prime(G) & = & \frac{2}{\pi}\,\int\limits_0^\infty \ln \left| \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}} { \sum\limits_{k=0}^{n-1} c_k\,(ix)^{n-k}} \right| dx \geq \frac{2}{\pi}\,\int\limits_0^\infty \ln Re \left[ \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}} { \sum\limits_{k=0}^{n-1} c_k\,(ix)^{n-k}} \right] dx \\[5mm] & = & \frac{2}{\pi}\,\int\limits_0^\infty \ln \left[ 1 + \frac{(-1)^{n/2}\,c_n\,A(x)}{A(x)^2+B(x)^2} \right] dx > 0\,. \end{eqnarray*} Formula (6) makes it possible to directly compute the effect of nullity in the special case of \(\eta(G)=0\) and \(\eta(G^\prime)=1\), using only the spectrum of the graph \(G\). Numerical examples will be communicated at some later moment. A same kind of consideration for the case \(\eta(G)=0\) and \(\eta(G^\prime)=2\), leads to \[ E(G)-E^\prime(G) = \frac{2}{\pi}\,\int\limits_0^\infty \ln \left| \frac{ \sum\limits_{k=0}^n c_k\,(ix)^{n-k}} { \sum\limits_{k=0}^{n-2} c_k\,(ix)^{n-k}}\right| dx, \] i.e., \[ E(G)-E^\prime(G) = \frac{2}{\pi}\,\int\limits_0^\infty \ln \left| \frac{\phi(G,ix)}{\phi(G,ix)-c_n-c_{n-1}\,ix}\right| dx\,. \] Other, more complicated cases can be treated in an analogous manner.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

Concepts, notation and graph parameters without formal definitions can be clarified in [1,2,3]. Unless stated otherwise, graphs will be finite, undirected and connected simple graphs. A shortest path having end vertices \(u\) and \(v\) is denoted by \(u-v_{(in~G)}\). If \(d_G(u,v)\geq 2\) then a vertex \(w\) on \(u-v_{(in~G)}\), \(w\neq u\), \(w\neq v\) is called an internal vertex on \(u-v_{(in~G)}\). When the context is clear the notation such as \(d_G(u,v)\), \(deg_G(v)\) can be abbreviated to \(d(u,v)\), \(deg(v)\) and so on.

Numerous graph or element parameters have been studied over years. These parameters can be categorized into two main groups. These are (i) graph structural properties such as, vertex degree, open and closed neighborhoods, graph diameter, connectivity, independence, domination and so on and (ii) derivative parameters stemming from a variety of vertex and/or edge labeling regimes or measure conditions. The latter is the study of the existence of vertex and/or edge subsets which establish graphical compliance with the definition of the stated labeling regime or measure condition.

Let \(\rho\) denote some minimum or maximum graph parameter related to subsets \(V(G)\) of graph \(G\). A minimum dominating set \(X\subset V(G)\) (therefore \(\rho=\gamma(G)=|X|\)) serves as an example. Let \(X\), \(Y\) be distinct subsets of \(V(G)\) which satisfy \(\rho\). Then \(X\) and \(Y\) is said to be parametric equivalent or \(\rho\)-equivalent denoted by, \(X\equiv_\rho Y\). Furthermore, if \(X\equiv_\rho Y\) and the induced graphs \(\langle V(G)\backslash X\rangle\), \(\langle V(G)\backslash Y\rangle\) are isomorphic then \(X\) and \(Y\) are said to be parametric isomorphic. This isomorphic relation is denoted by \(X \cong_\rho Y\). Let all the vertex subsets of graph \(G\) which satisfy \(\rho\) be \(X_1, X_2, X_3,\dots, X_k\). If \(X_1 \cong_\rho X_2 \cong_\rho X_3 \cong_\rho \cdots \cong_\rho X_k\) then \(X_i\), \(1\leq i \leq k\) are said to be parametric unique or \(\rho\)-unique. The graph \(G\) is said to have a parametric unique or \(\rho\)-unique solution (or parametric unique \(\rho\)-set). An interesting interpretation is that if \(G\) has a unique (exactly one) \(\rho\)-set \(X\), then \(X\) is a parametric unique \(\rho\)-set. The converse does not necessarily hold true. The notion of uniqueness has now been generalized for graph parameter studies.

The motivation is that in many real life application the choice between \(\rho\)-sets are often subject to additional conditions such as centrality, accessibility, domination, conflict with regards to transportation or data flow or clustering within social networks.

2. Confluence in graphs

The notion of a confluence set (a subset of vertices) of a graph \(G\) was introduced in [4]. For a non-complete graph \(G\), a non-empty subset \(\mathcal{X}\subseteq V(G)\) is said to be a confluence set if for every unordered pair \(\{u,v\}\) of distinct vertices (if such exist) in \(V(G)\backslash \mathcal{X}\) for which \(d_G(u,v)\geq 2\) there exists at least one \(u-v_{(in~G)}\) with at least one internal vertex, \(w\in \mathcal{X}\). Also any vertex \(u\in \mathcal{X}\) is called a confluence vertex of \(G\). A minimal confluence set \(\mathcal{X}\) (also called a \(\zeta\)-set) has no proper subset which is a confluence set of \(G\). The cardinality of a minimum confluence set is called the confluence number of \(G\) and is denoted by \(\zeta(G)\). A minimal confluence set is denoted by \(\mathcal{C}\). To distinguish between different graphs the notation \(\mathcal{C}_G\) may be used for a minimum confluence set of \(G\). Recall two important results from [4].

Theorem 1. [4] For a path \(P_n\), \(n\geq 1\), \begin{equation*} \zeta(P_n) = \begin{cases} 0, & \mbox{if \(n= 1\) or \(2\)};\\ \lfloor \frac{n}{3}\rfloor, & \mbox{if \(n \geq 3\)}.\\ \end{cases} \end{equation*}

Theorem 2. [4] For a cycle \(C_n\), \(n\geq 3\), \begin{equation*} \zeta(C_n) = \begin{cases} 0, & \mbox{if \(n=3\)};\\ 1, & \mbox{if \(n=4\)};\\ \lceil\frac{n}{3}\rceil, & \mbox{if \(n \geq 5\)}.\\ \end{cases} \end{equation*} The path \(P_3= v_1v_2v_3\) has confluence sets \(X_1=\{v_1\}\), \(X_2=\{v_2\}\), \(X_3=\{v_3\}\). Hence, \(X_1\equiv_\zeta X_2 \equiv_\zeta X_3\). Clearly, \(X_1 \cong_\zeta X_3\) because \(\langle (V(P_3)\backslash \{v_1\}\rangle \cong \langle V(P_3)\backslash \{v_3\}\rangle\). However, since \(X_1 \ncong_\zeta X_2 \ncong_\zeta X_3\) the path \(P_3\) does not have a parametric unique \(\zeta\)-set. Similar reasoning shows that path \(P_5 =v_1v_2v_3v_4v_5\) has a parametric unique \(\zeta\)-set. From [5] it is known that a path \(P_n\), \(n=5+3i\), \(i=0,1,2,\dots\) has a unique \(\zeta\)-set. Therefore, it has a parametric unique \(\zeta\)-set.

Proposition 1. A path \(P_n\) has a parametric unique \(\zeta\)-set if and only if \(n=1,2\) or \(n=4+3i\) or \(n=5+3i\), \(i=0,1,2,\dots\).

Proof. Part 1. The cases \(n=1,2\) follow from the fact that both \(P_1\), \(P_2\) are complete. Thus \(\mathcal{C}_{P_1}= \mathcal{C}_{P_2}=\emptyset\) and is respectively, unique. Therefore parametric unique.

For \(n=4+3i\), \(i=0,1,2,\dots\), the result follows from the fact that, without loss of generality, the \(\zeta\)-sets \(X_1=\{v_3, v_6,v_9,\dots,v_{n-1}\}\), \(X_2=\{v_3, v_6,v_9,\dots,v_{n-4},v_{n-2}\}\), \(X_3=\{v_3, v_6,v_9,\dots v_{n-7},v_{n-5},v_{n-2}\}\) and so on through back stepping until \(X_{\zeta(P_n)+1}=\{v_2, v_5,v_8,\dots v_{n-8},v_{n-5},v_{n-2}\}\), are all parametric isomorphic.

For \(n=5+3i\), \(i=0,1,2,\dots\), the result follows from the fact that \(P_n\) has a unique \(\zeta\)-set [5].

Part 2. If \(P_n\) has a parametric unique \(\zeta\)-set we use elimination through induction to prove the converse. It is easy to verify that \(P_n\) could be \(P_1\) or \(P_2\). For more valid converse options it is easy to verify that \(P_n\) could be \(n=4+3i\), \(n=5+3i\), \(i=0,1,2,\dots\). It is easy to verify that \(P_3\) does not have a parametric unique \(\zeta\)-set. For \(P_6\), \(\zeta(P_6)=2\). Obviously and amongst others, the sets \(X_1=\{v_3,v_5\}\) and \(X_2=\{v_3,v_6\}\) are \(\zeta\)-sets. Since, \(X_1 \ncong_\zeta X_2\) the converse does not hold for \(P_6\). Through immediate induction it follows that the converse does not hold for \(P_{3i}\), \(i=2,\dots\). Since, \(\mathbb{N}\backslash \{x \in \mathbb{N}: x=3i, i=1,2,\dots\} = \{y \in \mathbb{N}: y= 4+3i\), \(i=0,1,2,\dots\}\cup \{y \in \mathbb{N}: y= 5+3i\), \(i=0,1,2,\dots\}\cup \{1,2\}\) the converse follows.

Let a cycle \(C_n\), \(n\geq 3\) have vertex set \(V(C_n)=\{v_i: i=1,2,3,\dots,v_n\}\) and edge set \(E(C_n)=\{v_iv_{i+1}:1\leq i\leq n-1\}\cup \{v_1v_n\}\).

Proposition 2. A cycle \(C_n\) has a parametric unique \(\zeta\)-set if and only if \(n=3,4\) or \(n=5+3i\) or \(n=6+3i\), \(i=0,1,2,\dots\).

Proof. Part 1. The case \(n=3\) follows from the fact that \(C_3\) is complete. For \(n=4\) we have \(\{v_i\}\), \(1\leq i \leq 4\) a \(\zeta\)-set and \(\{v_i\}\cong_\zeta \{v_j\}\).

For \(n=5+3i\), \(i=0,1,2,\dots\), a valid minimum confluence set selection procedure is as follows. Let \(X_1=\{v_1,v_4,v_7,\dots,v_{n-1}\}\). It is easy to verify without loss of generality, that with \(v_1\) the initiation vertex the procedure yields a unique \(\zeta\)-set. Applying the procedure through consecutive modular counting for each \(v_i\), \(1\leq i\leq n\) yields unique sets \(X_i\) in respect of the initiation vertex \(v_i\). For each set \(X_i\) it follows that \[\langle V(C_n)\backslash X_i\rangle = \underbrace{P_2\cup P_2\cup\cdots \cup P_2}_{\lfloor \frac{n}{3}\rfloor~times} \cup P_1 = \lfloor\frac{n}{3}\rfloor P_2 \cup P_1.\] Hence, \(X_i\cong_\zeta X_j\) for all pairs of \(\zeta\)-sets. In fact the modular counting results in some identical \(\zeta\)-sets which need not be replicated. We conclude that cycles \(C_n\), \(n=5+3i\), \(i=0,1,2,\dots\) have parametric unique \(\zeta\)-sets.

For \(n=6+3i\), \(i=0,1,2,\dots\) the reasoning is similar.

Part 2. If \(C_n\) has a parametric unique \(\zeta\)-set we use elimination through induction to prove the converse. It is easy to verify that \(C_n\) could be \(C_3\) or \(C_4\). For more valid converse options it is easy to verify that \(C_n\) could be \(n=5+3i\), \(i=0,1,2,\dots\) or \(n=6+3i\), \(i=0,1,2,\dots\). For \(C_n\), \(n=7+3i\), \(i=0,1,2,\dots\) we note that \(C_7\) has at least the \(\zeta\)-sets, \(X_1=\{v_1,v_4,v_7\}\) and \(X_2=\{v_1,v_4, v_6\}\). Also, \(\langle V(C_7)\backslash X_1\rangle \ncong \langle V(C_7)\backslash X_2\rangle\) hence, \(X_1\ncong_\zeta X_2\). Therefore the converse does not hold for \(C_7\). Through immediate induction it follows that the converse does not hold for \(C_{(7+3i)}\), \(i=0,1,2,\dots\). Since, \(\mathbb{N}_{\geq 3}\backslash \{x \in \mathbb{N}: x=7 +3i\), \(i=0,1,2,\dots\} = \{y \in \mathbb{N}: y= 5+3i\), \(i=0,1,2,\dots\} \cup \{z \in \mathbb{N}: z= 6+3i\), \(i=0,1,2,\dots\} \cup \{3,4\}\) the converse follows.

Corollary 1. For a cycle \(C_n\) which has a parametric unique \(\zeta\)-set and with regards to vertex labeling let \(t\) be the number of parametric isomorphic \(\zeta\)-sets which originates from \(v_i\in V(C_n)\). Then \(C_n\) has \(\kappa(C_n)=\frac{nt}{\zeta(C_n)}\) distinct parametric isomorphic \(\zeta\)-sets. This implies that

• (a)   If \(C_n\), \(n= 6+3i\), \(i=0,1,2,\dots\) then \(\kappa(C_n)= \frac{n}{\zeta(C_n)}\).
• (b)   If \(C_n\), \(n= 5+3i\), \(i=0,1,2,\dots\) then \(\kappa(C_n)= n\).

Proof. The claim \(\kappa(C_n)=\frac{nt}{\zeta(C_n)}\) is trivial.

• (a)   Let \(n=6+3i\), \(i=0,1,2,\dots\): Thus \(\zeta(C_n)\geq 2\). From the proof of Proposition 2(Part 1) it follows that a vertex \(v_i \in V(C_n)\) yields exactly one \(\zeta\)-set. Hence, \(t=1\) implying that any two vertices in a \(\zeta\)-set initiate identical \(\zeta\)-sets. The aforesaid settles the result.
• (b)   Let \(n=5+3i\), \(i=0,1,2,\dots\): A valid minimum confluence set selection procedure is as follows. Let \(X_1=\{v_1,v_4, v_7,\dots, v_{n-1}\}\). It follows that \(X_2=\{v_1,v_4,v_7,\dots, v_{n-2}\}\) is valid. By similar modular back shifting it follows that \(X_3= \{v_1,v_4,v_7,\dots,v_{n-5},v_{n-2}\}\) is valid and so on. Hence, vertex \(v_1\) initiates exactly \(\zeta(C_n)\) parametric isomorphic \(\zeta\)-sets. Therefore, \(\kappa(C_n)=\frac{n\zeta(C_n)}{\zeta(C_n}=n\).

3. Types of trees

The authors are not aware of a unified classification of trees. The classification below is not a partition hence some categories (or families) are sub-categories of others. It is merely the specialization of structure which motivates the classification. When \(k\), \(k\geq 1\) leafs (or pendent vertices) are attached to a selected vertex \(v\) it is said that a \(k\)-bunch of leafs has attached to \(v\).

• (a)   Paths is a tree with exactly two leafs.
• (b)   A star \(S_{1,n}\), \(n\geq 3\) (sub-category of spiders) has a central vertex \(v_0\) with \(n\) leafs.
• (c)   A \(\ell\)-star \(S_{\ell,n\star m}\), \(\ell \geq 2\), \(n,m\geq 2\) has a path \(P_\ell =v_1v_2v_3\cdots v_\ell\) with a \(n\)-bunch of leafs attached to say, \(v_1\) and a \(m\)-bunch of leafs attached to \(v_\ell\).
• (d)   A spider \(S^\ast_n\), \(n\geq 3\) is a starlike tree with one vertex \(v_0\) of degree \(n\) and all other vertices have degree at most \(2\). Clearly, \(S^\ast_n\), \(n\geq 3\) has \(n\) pendent vertices. Hence in this context \(n\) does not mean the order of a spider. Put differently, a spider has a central vertex \(v_0\) which is attached with an edge to exactly one end-vertex of each path \(P_{m_1}, P_{m_2},\dots, P_{m_n}\), \(n\geq 3\).
• (e)   A caterpillar \(C_{n_1\star n_2\star,\cdots \star n_k}\), \(n_i,k\geq 1\) has a central path (or spine) \(P_m\), \(m\geq max\{3,k\}\) with each \(n_i\)-bunch of leafs, \(i=1,2,3,\dots, k\) attached to a distinct vertex of \(P_m\). If all \(n_i=1\) a trivial lobster is obtained.
• (f)   A lobster \(L_{T_1,T_2,T_3,\dots,T_\ell}\), \(T_i\in\{P_1,P_2,P_3, S_{1,n}\}\), \(\ell\geq 1\), has a central path \(P_m\), \(m\geq 1\) and the central vertex of each \(T_i\) be it an isolated vertex for \(P_1\) or an end-vertex for \(P_2\) or \(v_2\) for \(P_3\) or \(v_0\) for stars are connected by an edge to some vertex of \(P_m\). Hence a vertex of \(T_i\) is within distance 2 from some vertex of \(P_m\). A lobster has the property that if all leafs are removed a caterpillar is obtained.
• (g)   A finite \((k,d)\)-regular tree \(T_{k,d}\), \(d\geq 3\), \(k\geq 1\) are obtained as follows: Take a central vertex \(v_0\), \((\Rightarrow k=0)\) and attach \(d\) leafs to \(v_0\) (\(1^{st}\)-iteration \(\Rightarrow k=1\)), then for each new leaf attach \(d-1\) leafs (\(2^{nd}\)-iteration \(\Rightarrow k=2\)), then for each new leaf attach \(d-1\) leafs, \(\cdots\), until the \(k^{th}\)-iteration. Note that for a \((k,d)\)-regular tree \(T_{k,d}\), \(d\geq 3\), \(k\geq 1\) each leaf is an end-vertex of some \(diam\)-path and \(diam(T_{k,d})= 2k\) and \(v_0\) is on every \(diam\)-path.
• (h)   All other trees not in (a) through to (g).

Recall that the pendent degree of vertex \(u\in V(G)\) denoted by \(deg_p(u)\) be the number of leafs adjacent to \(u\). The open and closed pendent neighborhood of a vertex \(v\) are respectively, \(N_p(v)=\{\)leafs of \(v\}\) and \(N_p[v]=N_p(v)\cup \{v\}\). Also, a vertex \(v\) to which a leaf \(u\) is attached is called the pre-leaf of \(u\) or simply, pre-leaf \(v\). For paths the result with regards to parametric uniqueness is known (Proposition 1). A star has the unique \(\zeta\)-set, \(\{v_0\}\) which implies it has a parametric unique \(\zeta\)-set.

Proposition 3. A \(\ell\)-star \(S_{\ell,n\star m}\), \(\ell \geq 2\), \(n,m\geq 2\) has a parametric unique \(\zeta\)-set if and only if \(P_{\ell-2}=v_2v_3v_4\cdots v_{\ell-1}\), has \(\ell-2=4+3i\) or \(\ell-2=5+3i\), \(i=0,1,2,\dots\).

Proof. If \(\ell=2\), then \(\ell-2=0\), thus \(P_{\ell-2}=\emptyset\). For both \(\ell=3,4\) the paths \(P_1\), \(P_2\) are complete. For \(\ell-2=4+3i\) or \(\ell-2=5+3i\), \(i=0,1,2,\dots\), the result is a direct consequence of the proof found in Proposition 1 and the fact that both \(\langle N_p[v_2]\rangle\), \(\langle N_p[v_{\ell-1}]\rangle\) are stars.

Proposition 4. A spider \(S^\ast_n\), \(n\geq 3\) has a parametric unique \(\zeta\)-set if and only if, either (a) each \(P_{m_i}\), \(1\leq i\leq n\) has a parametric \(\zeta\)-set or (b) each \(P_{m_i}\), \(m_i=3j\), \(j=1,2,3,\dots\). Furthermore,

• (a)   \(\zeta(S^\ast_n)=1+\sum\limits_{i=1}^{n}\zeta(P_{m_i})\).
• (b)   \(\zeta(S^\ast_n)=\sum\limits_{i=1}^{n}\zeta(P_{m_i})\), \(m_i=3j\), \(j=1,2,3,\dots\).

Proof. It is known that a leaf need not be in a \(\zeta\)-set of any graph. See Lemma 8 in [5]. From the proof of Proposition 1 it follows that if a path has a parametric unique \(\zeta\)-set, an end-vertex cannot be in such set. Therefore \(v_0\) is in all \(\zeta\)-sets of \(S^\ast_n\) if each \(P_{m_i}\), \(1\leq i\leq n\) has a parametric \(\zeta\)-set. Hence, if all paths have a parametric unique \(\zeta\)-set then the spider has same. Hence, \(\zeta(S^\ast_n)=1+\sum\limits_{i=1}^{n}\zeta(P_{m_i})\).

If \(n\) copies of \(P_3\) are connected to \(v_0\) the \(\zeta\)-set is unique of order \(n\). Therefore the \(\zeta\)-set is parametric unique. Also \(v_0\) is not in the \(\zeta\)-set. However, \(P_3\) does not have a parametric unique \(\zeta\)-set. If some of the \(3\)-paths are substituted with paths of the form \(P_{m_i}\), \(m_i=3j\), \(j=2,3,\dots\), it follows easily through immediate induction that the \(\zeta\)-set is parametric unique and does not contain vertex \(v_0\). Hence, \(\zeta(S^\ast_n)=\sum\limits_{i=1}^{n}\zeta(P_{m_i})\), \(m_i=3j\), \(j=1,2,3,\dots\).

For paths \(P_{m_i}\), \(m_i=4+3j\), or \(m_i=5+3j\), \(j=0,1,2\dots\) the converse follows implicitly. For paths \(P_{m_i}\), \(m_i=3j\), \(j=1,2,3\dots\) the exclusion of \(v_0\) due to minimization results in a unique choice of a \(\zeta\)-set with regards to the paths.

Claim 1. For caterpillars \(C_{n_1\star n_2\star,\cdots \star n_k}\), \(n_i,k\geq 1\) a heuristic procedure will be described. This heuristic is adapted from the heuristic described for trees in [5].

• Step 1. Let \(X_1=\{v:v\in V(P_m), deg_p(v)\geq 2\}\). Delete all \(N_p[v]\), \(v\in X_1\) from the caterpillar.
• Step 2. Repeat Step 1 exhaustively to obtain sets \(X_2, X_3,\dots, X_t\). This is always possible and yields an explicit disconnected graph consisting of say, \(q\) components which could be paths and/or trivial caterpillars.
• Step 3. Utilize the heuristic for trees to obtain a \(\zeta\)-set of each of the \(q\) components. Label as sets \(Y_i\), \(1\leq i \leq q\).
• Step 4. If each \(Y_i\) is parametric unique then \(\mathcal{C}= [\bigcup\limits_{i=1}^{t}X_i] \cup [\bigcup\limits_{j=1}^{q}Y_j]\) is the parametric unique \(\zeta\)-set of the caterpillar. Otherwise, the caterpillar does not have a parametric unique \(\zeta\)-set.

Claim 2. For a lobster \(L_{T_1,T_2,T_3,\dots,T_\ell}\), \(T_i\in\{P_1,P_2,P_3, S_{1,n}\}\), \(\ell\geq 1\), a heuristic is described.

• Step 1. Let \(Z_1=\{v:v\in V(L_{T_1,T_2,T_3,\dots,T_\ell}), deg_p(v)\geq 2\}\). Delete all \(N_p[v]\), \(v\in Z_1\) from the lobster. The say, \(q\) components are path(s) and/or caterpillar(s).
• Step 2. Utilize the heuristic for caterpillars to obtain a \(\zeta\)-set of each of the \(q\) components. Label as sets \(Y_i\), \(1\leq i \leq q\).
• Step 3. If each \(Y_i\) is parametric unique then \(\mathcal{C}= [\bigcup\limits_{i=1}^{t}Z_i] \cup [\bigcup\limits_{j=1}^{q}Y_j]\) is the parametric unique \(\zeta\)-set of the lobster. Otherwise, the lobster does not have a parametric unique \(\zeta\)-set.

Proposition 5. A finite \((k,d)\)-regular tree \(T_{k,d}\), \(d\geq 3\), \(k\geq 1\) has a parametric unique \(\zeta\)-set and,

• (a)   If \(\ell \geq 2\) is even then, \(\zeta(T_{k,d})= d[1+\sum\limits_{t=3,5,7,\dots,(k-1)}(d-1)^{t-1}]\).
• (b)   If \(\ell \geq 3\) is odd then, \(\zeta(T_{k,d})= 1+d[\sum\limits_{t=2,4,6,\dots,(k-1)}(d-1)^{t-1}]\).

Proof. It follows easily that for \(k=0\) only \(v_0\) exists. For \(\ell \geq 1\), \(d \geq 3\) we have, level \(k=1 \Rightarrow d\) leafs, level \(k=2\Rightarrow d(d-1)\) leafs, \(\dots,\) level \(k=\ell \Rightarrow d(d-1)^{\ell-1}\) leafs.

• Case 1. If \(\ell \geq 2\) is even then \(\mathcal{C}=\bigcup\limits_{k=1,3,5,\dots,(k-1)}\{v:\) all \(v\) in level \(k\}\). It is easy to verify that \(\mathcal{C}\) is unique hence, parametric unique. Furthermore, \(\zeta(T_{k,d})= d[1+\sum\limits_{t=3,5,7,\dots,(k-1)}(d-1)^{t-1}]\).
• Case 2. If \(\ell \geq 3\) is odd then \(\mathcal{C}= \{v_0\}\cup \bigcup\limits_{k=2,4,6,\dots,(k-1)}\{v:\) all \(v\) in level \(k\}\). It is easy to verify that \(\mathcal{C}\) is unique hence, parametric unique. Furthermore, \(\zeta(T_{k,d})= 1+d[\sum\limits_{t=2,4,6,\dots,(k-1)}(d-1)^{t-1}]\).

4. Conclusion

In Corollary 1 the parameter \(\kappa(C_n)\) was introduced. From the proof of Proposition 1 it follows directly that for \(P_n\), \(n=4+3i\), \(i=0,1,2,\dots\) we have \(\kappa(P_n)= \zeta(P_n)+1= \lfloor\frac{n}{3}\rfloor +1\). For \(P_n\), \(n=5+3i\), \(i=0,1,2,\dots\) we have \(\kappa(P_n)= 1\). Furthering research for the parameter \(\kappa(G)\) in general remains open. Finding a \(\zeta\)-set for \(G\) in general is known to be NP-complete. Determining isomorphism between graphs is at least in the P-domain.

Conjecture 1. Consider any \(\zeta\)-set \(\mathcal{C}\) of graph \(G\), \(\zeta(G)\geq 2\). If for all \(\zeta\)-sets \(\mathcal{C}'\) derived from \(\mathcal{C}\) such that \(\mathcal{C}\cap \mathcal{C}'\neq \emptyset\) we have that \(\mathcal{C} \cong_\zeta \mathcal{C}'\) then \(G\) has a parametric unique \(\zeta\)-set.

If the conjecture is proven to be valid it opens an avenue to develop an efficient algorithm to determine parametric uniqueness in a graph. The principles are; (a) determine a \(\zeta\)-set where-after, (b) find all derivative \(\zeta\)-sets by substituting say \(u\in \mathcal{C}(G)\) with \(v\in N_2(u)\) and verifying confluence as well as parametric isomorphism. If the condition of confluence and parametric isomorphism are affirmative then parametric uniqueness follows. If the next stronger conjecture holds then the efficiency of an algorithm can be improved significantly.

Conjecture 2. Consider any \(\zeta\)-set \(\mathcal{C}\) of graph \(G\), \(\zeta(G)\geq 2\). Let \(v\in \mathcal{C}\). If for all \(\zeta\)-sets \(\mathcal{C}'\) derived from \(\mathcal{C}\) such that \(v \in\mathcal{C}\cap \mathcal{C}'\), we have that \(\mathcal{C} \cong_\zeta \mathcal{C}'\) then \(G\) has a parametric unique \(\zeta\)-set.

Researching parametric uniqueness with regards to confluence remains open for a vast range of interesting graph families such as circulants, cycle related graphs such as, wheel graphs, helm graphs, sunlet graphs, sun graphs and so on. Other graph parametric sets such as dominating sets, independent sets and others can be studied in respect of parametric uniqueness.

Acknowledgments

The authors would like to thank the anonymous referees for their constructive comments, which helped to improve on the elegance of this paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Cameron [1] considers non-negative lattice paths with an up-step of one unit and down-steps of \(1,3,5,\dots\) units. She gives (loc. cit.) credits to Emeric Deutsch for inventing these paths. Before we engage into further analysis of this situation (in a future publication), we will consider here the more modest model of down-steps of \(1,2,3,4,\dots\) units. Lattice paths with various step sizes have been thoroughly studied by Banderier and Flajolet [2], but never for an infinite number of possible steps.

We will call these paths Deutsch paths\(^{1}\), and require that they are non-negative and finish on a prescribed level \(i\). If \(i=0\), which is the analogue of Dyck paths, we talk about a closed Deutsch path. Since the up-steps and down-steps are not symmetrical, we can also investigate the model where the paths develop from right to left. In this case we talk about reversed Deutsch paths.

Dyck paths are extremely popular in combinatorics; the Encyclopedia of Integer Sequences [3] produces over 1200 hits. We restrict ourselves here to three additional references, all related to bijective ideas, [4,5,6].

Somewhat surprisingly, the computations that we will perform, lead us the world of Motzkin paths. Recall that the generating function of Motzkin paths (up-step, level-step, down-step) leads to \(M=1+zM+z^2M^2\), or

\begin{equation*} M(z)=\frac{1-z-\sqrt{1-2z-3z^2}}{2z^2}=1+z+2\,{z}^{2}+4\,{z}^{3}+9\,{z}^{4}+21\,{z}^{5}+51\,{z}^{6}+\cdots . \end{equation*} Here, as first shown in [7], the substitution \(z=\dfrac{v}{1+v+v^2}\) is beneficial: \begin{equation*} M=1+v+v^2,\quad\text{and}\quad v=\frac{1-z-\sqrt{1-2z-3z^2}}{2z}. \end{equation*} Introducing trinomial coefficients via \begin{equation*} \binom{n,3}{k}:=[v^k](1+v+v^2)^n \end{equation*} (notation is from Comtet [8]), we can compute the Motzkin numbers \begin{align*} [z^n]M(z)&=\frac1{2\pi i}\oint \frac{dz}{z^{n+1}}M(z)\\* &=\frac1{2\pi i}\oint \frac{dv}{v^{n+1}}(1-v^2)(1+v+v^2)^{n}\\* &=\binom{n,3}{n}-\binom{n,3}{n-2}. \end{align*} For the analysis of Deutsch paths, we will see that the same substitution works extremely well, and explicit answers can be given in terms of trinomial coefficients. The last section contains a bijection between open Deutsch paths (level at the end arbitrary) and Motzkin paths, of the same length.

2. Enumeration of Deutsch paths

We introduce an upper boundary \(h\), so that the paths live in a strip. This has the advantage that generating functions can be described by a system of linear equations, and solving it can be done by Cramer's rule. This involves the computation of some determinants.

We present the example of \(h=4\), and the generating functions \(\varphi_i(z)\) describe bounded Deutsch paths ending at level \(i\).

\begin{equation*} \left(\begin{matrix} 1&-z&-z&-z&-z\\ -z& 1&-z&-z&-z\\ 0& -z& 1&-z&-z\\ 0& 0& -z& 1&-z\\ 0& 0& 0& -z& 1\\ \end{matrix}\right) \left(\begin{matrix} \varphi_0\\ \varphi_1\\ \varphi_2\\ \varphi_3\\ \varphi_4\\ \end{matrix}\right)= \left(\begin{matrix} 1\\ 0\\ 0\\ 0\\ 0\\ \end{matrix}\right). \end{equation*} The \(n\times n \) determinant \begin{equation*} D_n=\frac{(1+v)^{n-1}}{(1+v+v^2)^n}\frac{1-v^{n+2}}{1-v} \end{equation*} of this system is of importance. It will eventually turn out that \begin{equation*} D_{h+1}=\frac{(1+v)^{h}}{(1+v+v^2)^{h+1}}\frac{1-v^{h+3}}{1-v}, \end{equation*} which can be obtained from the recursion \begin{equation*} (1+v+v^2)^2D_{n+2}-(1+v+v^2)(1+v)^2D_{n+1}+v(1+v)^2D_{n}=0. \end{equation*} Applying Cramer's rule leads to \begin{align*} \varphi_i&=\frac{z^iD_{h-i}}{D_{h+1}}=\frac{v^i}{(1+v+v^2)^{i}} \frac{(1+v+v^2)^{h+1}}{(1+v)^{h}}\frac{1-v}{1-v^{h+3}}\frac{(1+v)^{h-i-1}}{(1+v+v^2)^{h-i}}\frac{1-v^{h-i+2}}{1-v}\\ &=\frac{v^i(1+v+v^2)}{(1+v)^{i+1} }\frac{1-v^{h-i+2}}{1-v^{h+3}}. \end{align*} Of special interest is the generating function of closed bounded Deutsch paths: \begin{align*} \varphi_0 &=\frac{1+v+v^2}{1+v }\frac{1-v^{h+2}}{1-v^{h+3}}. \end{align*} Taking the limit \(h\to\infty\) leads to the enumeration of closed Deutsch paths, as originally desired: \begin{align*} \varphi_0 &=\frac{1+v+v^2}{1+v }. \end{align*} We can read off the coefficients explicitly: \begin{align*} [z^n]\varphi_0&=\frac1{2\pi i}\oint \frac{dz}{z^{n+1}}\frac{1+v+v^2}{1+v }\\ &=\frac1{2\pi i}\oint \frac{dv}{v^{n+1}}(1-v)(1+v+v^2)^{n}\\ &=\binom{n,3}{n}-\binom{n,3}{n-1}. \end{align*} The enumeration of closed Deutsch paths of height \(\ge h\) \begin{align*} \frac{1+v+v^2}{1+v }&-\frac{1+v+v^2}{1+v }\frac{1-v^{h+1}}{1-v^{h+2}} =\frac{1+v+v^2}{1+v }\frac{v^{h+1}(1-v)}{1-v^{h+2}}\\ \end{align*} is of interest, and we will discuss asymptotics of the average height in a later section.

Here is the generating function of bounded Deutsch paths with (arbitrary) open end:

\begin{align*} \sum_{i=0}^h\varphi_i &=(1+v+v^2)\frac{1-v^{h+1}}{1-v^{h+3}}. \end{align*} In the limit \(h\to\infty\), this leads to \begin{align*} 1+v+v^2, \end{align*} which is the generating function of Motzkin paths, according to length. A combinatorial explanation is given in the last section.

3. Enumeration of reversed Deutsch paths

This enumeration is quite similar, only the matrix is transposed: \begin{equation*} \left(\begin{matrix} 1&-z&0&0&0\\ -z& 1&-z&0&0\\ -z& -z& 1&-z&0\\ -z& -z& -z& 1&-z\\ -z& -z& -z& -z& 1\\ \end{matrix}\right) \left(\begin{matrix} \psi_0\\ \psi_1\\ \psi_2\\ \psi_3\\ \psi_4\\ \end{matrix}\right)= \left(\begin{matrix} 1\\ 0\\ 0\\ 0\\ 0\\ \end{matrix}\right). \end{equation*} The determinant is the same as before, but the application of Cramer's rule is slightly more unpleasant, as we have to distinguish cases. We only present the final results; intermediate calculations are not really difficult, but require some concentration. \begin{equation*} \psi_0=\frac{1+v+v^2}{1+v}\frac{1-v^{h+2}}{1-v^{h+3}}, \end{equation*} and for \(i\ge1\): \begin{equation*} \psi_i=v(1+v+v^2)(1+v)^{i-2}\frac{1-v^{h+1-i}}{1-v^{h+3}} \end{equation*} \begin{equation*} \sum_{i=0}^h\psi_i=(1+v+v^2)(1+v)^h\frac{1-v}{1-v^{h+3}}, \end{equation*} and the limit of this for \(h\to\infty\): \begin{equation*} (1+v+v^2)(1-v). \end{equation*} The enumeration in terms of trinomial coefficients: \begin{equation*} \binom{n,3}{n}-\binom{n,3}{n-1}-\binom{n,3}{n-2}+\binom{n,3}{n-3}. \end{equation*}

4. The average height of closed and open Deutsch paths

It was already worked out what the generating function of closed Deutsch paths is, and for the average height, one needs to compute \begin{align*} \sum_{h\ge1}&\frac{1+v+v^2}{1+v }\frac{v^{h+1}(1-v)}{1-v^{h+2}}=-\frac{1+v+v^2}{1+v } -\frac{v(1+v+v^2)}{(1+v)^2 }+ \frac{(1+v+v^2)(1-v)}{v(1+v) }\sum_{h\ge1}\frac{v^{h}}{1-v^{h}}. \end{align*} The standard procedure is to find the local behaviour of this near the singularity \(v=1\leftrightarrow z=\frac13\). A fairly detailed description of this can be found in [9]. Somewhat similar computations will also appear in [10]. We get \begin{align*} E(z)&= \sum_{h\ge1}\frac{1+v+v^2}{1+v }\frac{v^{h+1}(1-v)}{1-v^{h+2}}\\ &=-\frac94 + \frac{3(1-v)}{2}\Big[-\frac{\log(1-v)}{1-v}+\frac{\gamma}{1-v}+\cdots\Big]\\ &=-\frac32\log(1-v)+\textsf{const.}+\cdots. \end{align*} Since \(1-v\sim\sqrt3\sqrt{1-3z}\), this leads further to the local expansion \begin{equation*} E(z)\sim-\frac34\log(1-3z)+\cdots. \end{equation*} Traditional singularity analysis [11] leads to \begin{equation*} [z^n]E(z)\sim \frac34\frac{3^n}{n}. \end{equation*} This has to be divided by the total number of closed Deutsch paths, with generating function \begin{equation*} \frac{1+v+v^2}{1+v}\sim 3-\frac{9}{4\sqrt3}\sqrt{1-3z}, \end{equation*} so that \begin{equation*} [z^n]\frac{1+v+v^2}{1+v}\sim \frac{9}{8\sqrt{3\pi}}3^nn^{-3/2}. \end{equation*} The quotients is the average height of closed Deutsch paths (leading term only): \begin{equation*} \frac{\frac34\frac{3^n}{n}}{\frac{9}{8\sqrt{3\pi}}3^nn^{-3/2}}=2\sqrt{\frac{\pi n}{3}}. \end{equation*} In the classical case of Motzkin paths [7], this height is \(\sqrt{\frac{\pi n}{3}}\), so Deutsch paths are about twice as high.

We would also like to do this type of analysis for open Deutsch paths, as they are exactly enumerated by Motzkin numbers. Recall the generating function

\begin{align*} \sum_{i=0}^h\varphi_i &=(1+v+v^2)\frac{1-v^{h+1}}{1-v^{h+3}} \end{align*} of open Deutsch paths, with height \(\le h\). Taking differences, we get the generating function of open Deutsch paths, with height \(\ge h\): \begin{equation*} (1+v+v^2)(1-v^2)\frac{v^h}{1-v^{h+2}}. \end{equation*} This has to be summed: \begin{equation*} (1+v+v^2)(1-v^2)\sum_{h\ge1}\frac{v^h}{1-v^{h+2}}\sim -6\log(1-v)\sim-3\log(1-3z), \end{equation*} with coefficients asymptotic to \(\frac{3^{n+1}}{n}\).

This has to be divided by

\begin{equation*} [z^n]M(z)\sim\frac{9}{2\sqrt{3\pi }}3^nn^{-3/2}; \end{equation*} which leads again to the average height \begin{equation*} 2\sqrt{\frac{\pi n}{3}}. \end{equation*} So open versus closed does not influence the (leading term of the) average height.

5. The LU-decomposition

We compute the LU-decomposition of the Deutsch matrix, where indices run from 1 to \(n\). This isn't strictly necessary, but always interesting, and leads to the determinant as a by-product. \begin{equation*} L_{i,i}=1, \end{equation*} \begin{equation*} L_{i,i-1}=-\frac{v}{1+v}\frac{1-v^i}{1-v^{i+1}}, \end{equation*} other values 0. \begin{equation*} U_{i,i}=\frac{1+v}{1+v+v^2}\frac{1-v^{i+2}}{1-v^{i+1}}, \end{equation*} \begin{equation*} U_{i,j}=-v\frac{1+v}{1+v+v^2}\frac{1-v^{i}}{1-v^{i+1}}. \end{equation*} For the conformation, we make this computation: \begin{align*} \sum_k L_{i,k}U_{k,j}&=U_{i,j}-\frac{v}{1+v}\frac{1-v^i}{1-v^{i+1}}U_{i-1,j}\\ &=-v\frac{1+v}{1+v+v^2}\frac{1-v^{i}}{1-v^{i+1}}+\frac{v}{1+v}\frac{1-v^i}{1-v^{i+1}} v\frac{1+v}{1+v+v^2}\frac{1-v^{i-1}}{1-v^{i}}\\ &=-\frac{v}{1+v+v^2}=-z. \end{align*} Now let us consider the special cases, where \(U_{i,i}\) appears: \begin{align*} \sum_k L_{i,k}U_{k,i}&=U_{i,i}-\frac{1+v}{1+v+v^2}\frac{1-v^{i+2}}{1-v^{i+1}}\\ &=\frac{1+v}{1+v+v^2}\frac{1-v^{i+2}}{1-v^{i+1}}+ \frac{v}{1+v}\frac{1-v^i}{1-v^{i+1}} v\frac{1+v}{1+v+v^2}\frac{1-v^{i-1}}{1-v^{i}}=1, \end{align*} \begin{align*} \sum_k L_{i,k}U_{k,i-1}&=U_{i,i-1}-\frac{v}{1+v}\frac{1-v^i}{1-v^{i+1}}U_{i-1,i-1}\\ &=-v\frac{1+v}{1+v+v^2}\frac{1-v^{i}}{1-v^{i+1}} -\frac{v}{1+v}\frac{1-v^i}{1-v^{i+1}} \frac{1+v}{1+v+v^2}\frac{1-v^{i+1}}{1-v^{i}}\\ &=-\frac{v}{1+v+v^2}=-z. \end{align*} In particular, we get for the determinant \begin{equation*} U_{1,1}\dots U_{n,n}=\frac{(1+v)^n}{(1+v+v^2)^n}\frac{1-v^{n+1}}{1-v^{2}}, \end{equation*} which checks with our previous observation.

Very briefly, we only cite the results of the transposed matrix (related to reversed Deutsch paths):

\begin{equation*} L_{i,i}=1 \end{equation*} and for \(j< i\) \begin{equation*} L_{i,j}=-v\frac{1-v^j}{1-v^{j+2}}. \end{equation*} Furthermore \begin{equation*} U_{i,i}=\frac{1+v}{1+v+v^2}\frac{1-v^{i+2}}{1-v^{i+1}}, \end{equation*} \begin{equation*} U_{i,i+1}=-\frac{v}{1+v+v^2}, \end{equation*} and zero for all other values of the \(U\)-matrix.

6. The total area

The area of a closed Deutsch path \(a_0a_1\dots a_n\) with \(a_0=a_n=0\) is defined to be \(a_0+\cdots+a_n\). With the generating functions that we worked out, the generating function of the total area, summed over all Deutsch paths of length \(n\), can be computed: \begin{align*} A(z)&=\sum_{i\ge1}i\cdot\varphi_i\cdot\psi_i=\sum_{i\ge1}i\frac{v^i(1+v+v^2)}{(1+v)^{i+1}}v(1+v+v^2)(1+v)^{i-2}= \frac{v^2(1+v+v^2)^2}{(1+v)^3(1-v)^2}\\ &={z}^{2}+3\,{z}^{3}+12\,{z}^{4}+39\,{z}^{5}+129\,{z}^{6}+411\,{z}^{7}+ 1300\,{z}^{8}+4065\,{z}^{9}+12633\,{z}^{10}+\cdots. \end{align*} We can also compute that \begin{equation*} A(z)\sim \frac 98\frac1{(1-v)^2}\sim \frac38\frac{1}{1-3z}, \end{equation*} so that \begin{equation*} [z^n]A(z)\sim \frac38 3^n. \end{equation*} Dividing this by the total number of closed Deutsch paths \begin{equation*} \frac{9}{8\sqrt{3\pi }}3^nn^{-3/2} \end{equation*} leads to \begin{equation*} \sqrt{\frac{\pi}{3}}n^{3/2}. \end{equation*} One might even divide this by \(n\), the length of the Deutsch paths, and can interpret \(\sqrt{\frac{\pi n}{3}}\) as the average elevation of a random closed Deutsch path of length \(n\). Notice that the average height is about twice this quantity.

7. A bijection

The generating function of Motzkin paths satisfies \(M=1+zM+z^2M^2\), which is based on a first return decomposition. We will now derive the same recursion for open Deutsch paths, which leads to a recursively defined bijection.

We will define a map \(\psi\) in a recursive way, mapping an open Deutsch path to a Motzkin path; it is easily seen to be reversible. Of course the empty path is mapped to the empty path. Now, if the open Deutsch path never returns to the \(x\)-axis, it can be described by \(w=U\widetilde{w}\), where \(U\) describes an up-step, and \(\widetilde{w}\) is itself an open Deutsch path. Then we map it to \(F\psi(\widetilde{w})\), where \(F\) is a flat step. Otherwise, if the open Deutsch path returns to the \(x\)-axis for the first time, it may be written as \(w=U\widetilde{w}Dx\), where \(D\) is a down-step of any size. Note that both, \(\widetilde{w}\) and \(x\) are themselves open Deutsch paths. Then we map this to \(U\psi(\widetilde{w})D\psi(x)\), which is a Motzkin path (here, \(D\) is just a down-step of one unit).

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

Adominating set of a graph \(G=(V,E) \) is a subset \(S\subseteq V \) such that every vertex is either adjacent to a vertex in \(S \) or is in \(S \) itself, i.e., \(V\setminus S\subseteq N(S) \), or, alternatively, \(S\cup N(S)=V \), where \(N(S) \) is the open neighborhood of \(S \). This notion can be extended to digraphs \(D=(V,A) \) through out-neighborhoods by finding a subset \(S\subseteq V \) such that every vertex is in either \(N^{+}(S) \) or in \(S \) itself, i.e., \(S\cup N^{+}(S)=V \), where \(N^{+}(S) \) is the open out-neighborhood of \(S \).

Domination and dominating set problems come in a variety of flavors, and date back at least to a chess problem found in [1] about which positions a queen can dominate on a chessboard. Since then, many applications of domination and dominating sets in graphs and networks have been discovered. Results on graph separability using dominating sets were obtained in [2]. Progress on binary locating sets for vertices in polytopes occurred in [3]. In [4] bounds were found for the dominator number of hypercube networks, in particular for Fibonacci cubes.

Characterizing these sets and their potential applications are interesting and important problems. For example, while every graph with no isolated vertex admits at least two disjoint dominating sets, this is not true in general for total dominating sets. In fact, a recent paper by Henning and Peterin [5] characterized graphs with two disjoint total dominating sets. But typical applications of dominating sets involve characterizing the vertices of dominating sets, not the dominating sets themselves. For this purpose, an interesting and relatively new concept was developed, that of dominator colorings.

A dominator coloring of a graph is a proper vertex coloring in which every vertex dominates at least one color class. The dominator chromatic number of a graph is the size of a minimum dominator coloring. Dominator colorings and dominator chromatic numbers of graphs were first introduced in papers by Dr. Gera [6,7]. Since that initial work, dominator colorings have been studied intensively in specific graph families including bipartite graphs [8], trees [9], and certain Cartesian products [10]. More general results were obtained in [11] and [12]. Algorithmic results were first obtained in [13] and [14] in a general setting as well as in [15] in which a specific algorithm for finding minimum dominator colorings of trees was developed. These results led to the vast development of applications of dominating sets in undirected networks, including, e.g., [16,17,18,19,20].

Recently the notion of dominator colorings was extended to directed graphs in [21]. In that paper the focus was on finding the dominator chromatic number over all possible orientations of paths and cycles. In this paper, for the most part, we shift our attention from this perspective towards finding the exact dominator chromatic number of specific structures. We note that in [22] a linear algorithm for finding the minimum dominator coloring of an oriented path was developed and open source software implementing this algorithm was released. Before continuing, we provide a formal definition of a dominator coloring for a directed graph.

Definition 1. A dominator coloring of a directed graph is a proper vertex coloring which further satisfies the requirement that every vertex dominates at least one color class in its out-neighborhood.

Notice the importance of orientation here, if we did not specify that the dominator coloring was with respect to the out-neighborhood, then this problem would be analogous to that of dominator colorings of undirected graphs. This means that the definition of domination in a directed graph can be expressed in terms of the ordered pairs of vertices that comprise the arcs; for an arc \(e=(u,v) \) we say that the vertex \(u \) dominates the vertex \(v \), but not vice versa.

All digraphs in this paper are orientations of simple, finite trees. One particular structure which will be a primary focal point in this paper is the arborescence. An arborescence, also known as an out-tree, is the orientation of a tree in which all arcs point away from a single source. Similarly, an anti-arborescence, or an in-tree, is the orientation of a tree in which all arcs point towards a single sink. As we will see, the dominator chromatic number of arborescences is invariant under reversal of orientation.

This rather interesting results naturally begs to be generalized. Surprisingly yet beautifully, the marquee result in this paper is that the dominator chromatic number of an oriented tree \(T \) is invariant under reversal for all oriented finite trees.

Before continuing this work, we declare several important notations. Unless otherwise specified, the cardinality of the vertex set \(V(T) \) of a given tree \(T \) is denoted by \(n \). The set of all leaves of \(T \) is denoted by \(l(T) \) and the number of leaves in a tree, i.e., \(|l(T)| \), is denoted by \(l \) when \(T \) is known. For an oriented tree \(T \), we will denote its reversal by \(T^{-} \). The out-degree of a vertex \(v \) is denoted by \(d^{+}(v) \). Finally, we will use \(\chi_{d}(D) \) to denote the dominator chromatic number of a given digraph \(D \).

2. Arborescences

In this section we study arborescences, building up to a proof of the dominator chromatic number of any arborescence or anti-arborescence. First, however, we recap previous results on orientations of paths.

The simplest type of tree is a path. It is easy to see that the dominator chromatic number of the directed path \(P_{n} \) is \(n \) and that this result is trivially invariant under reversal. The following theorem from [21] gives the dominator chromatic number over all orientations of paths.

Theorem 2. The minimum dominator chromatic number over all orientations of the path \(P_{n} \) is given by \begin{equation*} \chi_{d}(P_{n})=\begin{cases} k+2 & \mathrm{if}\ n=4k,\\ k+2 & \mathrm{if}\ n=4k+1,\\ k+3 & \mathrm{if}\ n=4k+2,\\ k+3 & \mathrm{if}\ n=4k+3, \end{cases} \end{equation*} for \(k\geq 1 \) with the exception \(\chi_{d}(P_{6})=3 \).

Moving on from paths, especially when considering directed paths specifically, arborescences (and anti-arborescences) can be considered an immediate extension. As we build towards our proof that the dominator chromatic number of (anti-)arborescences is invariant under reversal of orientation, we first characterize the dominator chromatic number of (anti-)arborescences in terms of their parameters, namely the size of their vertex set and the number of leaves.

Lemma 3. Let \(T \) be an in-tree (anti-arborescence). Then \(\chi_{d}(T)=n-l+1 \).

Proof. Since the set of leaves of \(T \) form an independent set and since \(d^{-}(v)=0 \) for all leaves \(v \) ( \(T \) is an in-tree), we may assign the set \(l(T) \) to the same color class. It remains to be shown that every non-leaf vertex of \(T \) must be uniquely colored. This follows from the fact that \(d^{+}(v)\leq 1 \) for all \(v\in V(T) \) when \(T \) is an in-tree, as this implies that each non-leaf vertex is the only vertex dominated by some other vertex in \(T \) and therefore must be uniquely colored.

Lemma 4. Let \(T \) be an out-tree (arborescence). Then \(\chi_{d}(T)=n-l+1 \).

Proof. The proof is by contradiction via minimum counterexample with respect to \(n=|V(T)| \). Since a single vertex \(v \) is technically an arborescence with one leaf and since \(\chi_{d}(v)=1 \), we may assume that some tree \(T \) is a minimum counterexample to our claim. Let \(|V(T)|=n \), let \(v \) be a leaf of \(T \) with in-neighbor \(u \), and let \(T^{\prime}=T\setminus\{v\} \). Since \(|V(T^{\prime})|< |V(T)| \), it follows that \(\chi_{d}(T^{\prime})=(n-1)-(l-1)+1=n-l+1 \) if \(v \) was not the only out-neighbor of \(u \), and \(\chi_{d}(T^{\prime})=(n-1)-l+1=n-l \) if \(v \) was the only out-neighbor of \(u \) in \(T \). If \(v \) was not the only out-neighbor of \(u \), then we may color \(v \) with the same color as the other out-neighbors of \(u \), and if \(v \) was the only out-neighbor of \(u \) then we color \(v \) uniquely. In either case we have established that \(\chi_{d}(T)=n-l+1 \) which contradicts our assumption that \(T \) was a minimum counterexample and conclude that \(\chi_{d}(T)=n-l+1 \) for any arborescence \(T \).

Corollary 5. For any (anti-)arborescence \(T \), we have \(\chi_{d}(T)=\chi_{d}(T^{-}) \) where \(T^{-} \) is \(T \) with the orientation of every arc reversed.

Notice that Theorem 2 is a special case of both of the above lemmas and their corollary.

3. Main Results

In this section we prove our main result, that the dominator chromatic number of an oriented tree is invariant under reversal. To do this we will prove several necessary lemmas en route. First, however, we begin with an observation.

Observation 1. Let \(T \) be an orientation of a tree and let \(v \) be a leaf of \(T \). Then \(\chi_{d}(T)-1\leq\chi_{d}(T\setminus\{v\})\leq\chi_{d}(T) \).

Proof. Clearly if the lower bound is false then we may find a smaller dominator chromatic number for \(T \). The upper bound is obvious.

Next, in order to prove our main theorem, we need a more refined analysis of the subtree \(T^{\prime}=T\setminus\{v\} \) for some leaf \(v \) of \(T \). We begin by first providing a characterization of the subtree \(T^{\prime} \) when \(\chi_{d}(T^{\prime})=\chi_{d}(T)-1 \).

Lemma 6. Let \(T \) be an orientation of a tree and let \(v \) be a leaf of \(T \) with neighbor \(u \). Then \(\chi_{d}(T\setminus\{v\})=\chi_{d}(T)-1 \) if and only if either \(\{v\}=N^{+}(u) \) or \(\{v\}=\{x\in V(T)|d^{-}(x)=0\} \).

Proof. (\(\impliedby\)) It is obvious that \(v \) must be uniquely colored in any minimum dominator coloring of \(T \) in either case.

(\(\implies\)) Since \(v \) is a leaf of \(T \), \(d^{+}(v)+d^{-}(v)=1 \). If \(d^{-}(v)=1 \) it suffices to show that if \(|N^{+}(u)|>1 \) then \(\chi_{d}(T\setminus\{v\})=\chi_{d}(T) \). This follows immediately by seeing that for any \(x\in N^{+}(u)\setminus\{v\} \) it must be that if \(v \) is uniquely colored, we may recolor \(v \) with \(c(x) \) thereby contradicting our assumption that our dominator coloring was a minimum dominator coloring. If \(d^{+}(v)=1 \) and there exists some other vertex \(x \) such that \(d^{-}(x)=0 \), then if \(v \) was uniquely colored, we may again color \(v \) with \(c(x) \) and contradict our assumption that our dominator coloring was a minimum dominator coloring of \(T \).

With that lemma intact, we further refine our understanding of the subtree \(T^{\prime}=T\setminus\{v\} \) by studying the neighbor of the leaf vertex \(v \).

Lemma 7. Let \(T \) be an orientation of a tree with leaf \(v \) satisfying \(\chi_{d}(T\setminus\{v\})=\chi_{d}(T)-1 \). If \(d^{-}(v)=0 \) then \(d^{-}(u)=1 \) where \(u \) is the neighbor of \(v \) in \(T \).

Proof. Clearly \(u \) cannot have any additional in-neighbors that are also leaves, else \(v \) would not be uniquely colored in any minimum dominator coloring of \(T \). Moreover, all leaves \(l\neq v \) of \(T \) must have \(d^{-}(l)=1 \), else \(\{v\}\neq\{x\in V(T)|d^{-}(x)=0\} \) in which case Lemma 6 tells us that \(\chi_{d}(T\setminus\{v\})\neq\chi_{d}(T)-1 \), a contradiction.

Let \(T^{\prime} \) be the subtree of \(T \) obtained by deleting all leaves of \(T \) except for \(v \). If \(T^{\prime} \) has a leaf \(l \) with \(d^{-}(l)=0 \) then the vertex \(l \) has \(d^{-}(l)=0 \) in \(T \). By iterating this process until \(T^{n}=vu \) we see that it must be the case that \(d^{-}(u)=1 \) in \(T \).

Finally we are ready to prove the main result of this paper, that the dominator chromatic number of an oriented tree is invariant under reversal.

Theorem 8. Let \(T \) be an orientation of a tree and let \(T^{-} \) be \(T \) with the orientation of every arc reversed. Then \(\chi_{d}(T)=\chi_{d}(T^{-}) \).

Proof. Let \(T \) be a minimum counterexample, let \(v \) be a leaf of \(T \) with neighbor \(u \), and let \(T^{\prime}=T\setminus\{v\} \). We prove this by considering the cases where \(\chi_{d}(T^{\prime})=\chi_{d}(T)-1 \) and where \(\chi_{d}(T^{\prime})=\chi_{d}(T) \). Within each case we will consider the two possible subcases, where \(d^{-}(v)=0 \) and where \(d^{+}(v)=0 \).

First, assume that \(\chi_{d}(T^{\prime})=\chi_{d}(T)-1 \). If \(d^{-}(v)=0 \) then by Lemma 6 we know that \(\{v\}=\{x\in V(T)|d^{-}(x)=0\} \). This implies that \(d^{-}(u)=1 \) by Lemma 7. Thus \(d^{+}_{T^{-}}(u)=1 \) and \(v \) must be uniquely colored and so \(\chi_{d}(T^{\prime})=\chi_{d}(T) \).

If \(d^{+}(v)=0 \) then \(d^{+}(u)=1 \) else \(v \) would not be uniquely colored in \(T \). Thus \(d^{-}(u)=1 \) in \(T^{\prime-} \). If \(u \) is not uniquely colored in \(T^{\prime-} \), then we may uniquely recolor \(u \) and assign to \(v \) the original color had by \(u \) in \(T^{\prime-} \). Since \(d^{-}(u)=0 \) in \(T^{\prime-} \), if there exists \(x\in V(T^{\prime-}) \) such that \(c(x)=c(u) \), then this color class is not dominated by any vertex. Furthermore, if there exists any non-dominated color classes in \(T^{\prime-} \) then \(u \) must belong to such a class, else our dominator coloring of \(T^{\prime-} \) is not minimal. Therefore, if \(u \) is uniquely colored in a minimum dominator coloring of \(T^{\prime-} \), there does not exist any non-dominated color class besides \(c(u) \) in \(T^{\prime-} \). This implies that \(\{u\}=\{x\in V(T^{\prime-})|d^{-}(u)=0\} \) which implies that \(v \) must be uniquely colored in order to have a proper dominator coloring of \(T^{-} \), hence \(\chi_{d}(T^{-})=\chi_{d}(T) \).

Now assume that \(\chi_{d}(T^{\prime})=\chi_{d}(T) \). If \(d^{+}(v)=0 \) then \(\exists\ x\in N^{+}(u) \) such that \(x\neq v \) since \(v \) is not uniquely colored in any minimum dominator coloring of \(T \). Since \(|V(T^{\prime})|< |V(T)| \) it follows that \(\chi_{d}(T^{\prime-})=\chi_{d}(T^{\prime})=\chi_{d}(T) \). Since \(xu\in A(T^{\prime}-) \) and \(d^{+}(x)=1 \) in \(T^{\prime}- \), it follows that \(u \) is uniquely colored in any minimum dominator coloring of \(T^{\prime}- \). We may then add \(v \) to \(T^{\prime-} \) and color it with \(c(x) \) to establish that \(\chi_{d}(T^{-})=\chi_{d}(T) \).

If \(d^{-}(v)=0 \) then \(\exists\ x\neq v \) such that \(d^{-}(x)=0 \) by Lemma 6. If such a vertex is also an in-neighbor of \(u \) in \(T \) then we may color \(v \) with \(c(x) \) and be done, so we may assume that \(\{v\}=N^{+}(u) \) in \(T^{\prime-} \). Since \(|V(T^{\prime})|< |V(T)| \), it follows that \(\chi_{d}(T^{\prime-})=\chi_{d}(T^{\prime}) \). Since \(\{v\}=N^{-}(u) \) in \(T \), it follows that \(d^{-}(u)=0 \) in \(T^{-} \). This mean that \(|\{w\in V(T^{\prime})|d^{-}(w)=0\}|\geq 2 \). If \(x \) is a leaf in \(T \), then \(x \) cannot be the only in-neighbor of its out-neighbor in \(T^{\prime} \), else \(x \) would be uniquely colored in any minimum dominator coloring of \(T^{\prime} \) and we could consider the tree \(T\setminus\{x\} \) which satisfies \(\chi_{d}(T\setminus\{x\})< \chi_{d}(T) \) and use the first case to complete the proof. Therefore we may assume that \(\chi_{d}(T^{\prime\prime-})=\chi_{d}(T^{\prime\prime}) \) where \(T^{\prime\prime}=T\setminus\{x\} \). Let \(w \) be the in-neighbor of \(x \) in \(T^{-} \). Since \(x \) is not the only member of \(N^{+}(w) \), \(w \) already dominates a color class in any minimum dominator coloring of \(T^{\prime\prime-} \), all one such color class \(\hat{c} \). By coloring \(x \) with \(\hat{c} \) in a given minimum dominator coloring of \(T^{-} \), we establish that if \(x \) is a leaf in \(T \), that \(\chi_{d}(T^{-})=\chi_{d}(T) \).

Finally, assume that there is no such leaf vertex in \(T \), i.e., that for all leaves \(l\neq v \) of \(T \), \(d^{-}(l)=1 \). If there exists some leaf \(l\neq v \) such that \(\{l\}=N^{+}(N^{-}(l)) \), the second subcase of the first case completes the proof. Therefore, we may assume that for every leaf \(l\neq v \) (which there necessarily exists at least one such leaf) we have that there exists some other leaf \(l^{\prime} \) such that \(l^{\prime} \) is also a member of \(N^{+}(N^{-}(l)) \). Let \(T^{\prime\prime}=T\setminus\{l\} \). Since \(|V(T^{\prime\prime})|< |V(T)| \), it follows that \(\chi_{d}(T^{\prime\prime-})=\chi_{d}(T^{\prime\prime}) \). We may color \(l \) with \(c(l^{\prime}) \) for some \(l^{\prime}\in N^{-}(N^{ +}(l)) \) in \(T^{-} \), establishing that \(\chi_{d}(T^{-})=\chi_{d}(T) \). As this concludes the last possible case, the proof is complete.

4. Applications

In this section we use our main result, that the dominator chromatic number of an orientation of a tree is invariant under reversal of orientation, as a tool to help prove results on other oft studied tree topologies including generalized stars and caterpillars.

We begin by studying orientations of generalized stars. In some sense a generalized star could be construed to mean a union of paths of varying lengths that all meet at a single common vertex. However, for the sake of this paper we consider generalized stars \(GS_{m}^{k} \) which consist of \(m \) paths consisting of \(k \) edges, all originating from a single, common vertex. Note that the traditional star graph \(S_{m} \) is also \(GS_{m}^{1} \). An illustration of \(GS_{8}^{2} \) is given below for reference.

The following result from [21] tells us the exact dominator chromatic number of a star for every possible orientation.

Proposition 9. Let \(D \) be an orientation of a star graph, \(G \). Then we have that \(2\leq\chi_{d}(D)\leq3 \), \(\chi_{d}(D)=2 \) if and only if all arcs are oriented similarly with respect to the central vertex, and \(\chi_{d}(D)=3 \) otherwise.

It is easy to see from this result that the dominator chromatic number of an oriented star is invariant under reversal. We can now attempt to expand this result by proving further results on the dominator chromatic number of generalized stars \(GS_{m}^{k} \).

Lemma 10. Let \(GS_{m}^{k} \) be an orientation of generalized star featuring either a single source or a single sink. Then \(\chi_{d}(GS_{m}^{k})=m(k-1)+2 \).

Proof. This follows from Lemmas 3 and 4 and from Theorem 8. Since \(GS_{m}^{k} \) has \(m \) paths extending from its central vertex, it has \(m \) leaves. Since \(|V(GS_{m}^{k})|=n=mk+1 \), it follows that \(\chi_{d}(GS_{m}^{k})=mk-m+2=m(k-1)+2 \).

A natural question to ask would be whether or not this is best possible. It turns out that this is not the case. We improve greatly on this bound with our next result.

Lemma 11. For \(k\geq 2 \) we have that the minimum dominator chromatic number of all possible orientations of the generalized star satisfies \(\chi_{d}(GS_{m}^{k})\leq 3+m\left(\left\lfloor\frac{k}{2}\right\rfloor-1\right) \).

Proof. The proof is by construction. Begin by considering \(GS_{m}^{2} \). Let \(S_{0} \) be the central vertex of \(GS_{m}^{2} \), let \(S_{1}=N(S_{0}) \), and let \(S_{2}=N(S_{1})\setminus S_{0} \). Notice that we have simply created independent sets of ``layers" of the generalized star \(GS_{m}^{2} \). We orient all arcs away from \(S_{1} \) into \(S_{0} \) and \(S_{2} \). We may assign a single color to \(S_{1} \) since no vertex has positive in-degree, and we may assign a single color to \(S_{0} \) as it is a single vertex. Since every vertex in \(S_{1} \) dominates, \(S_{0} \), we may color all of \(S_{2} \) with a third color, demonstrating that \(\chi_{d}(GS_{m}^{2})\leq 3 \).

We generalize this structure by coloring all vertices in each \(S_{2i+1} \) with the same color used on \(S_{1} \), by coloring all vertices in each \(S_{2i} \) with a unique color, and by orienting all arcs from \(S_{2i+1} \) to \(S_{2i} \). Notice that the union \(\bigcup S_{2i+1} \) constitutes an independent set comprised of vertices with in-degree equal to zero, hence they may all share the color assigned to \(S_{1} \). Since all vertices in \(\bigcup S_{2i} \) have out-degree zero and are uniquely colored, every vertex in \(\bigcup S_{2i+1} \) dominates some uniquely colored vertex. The result follows from basic counting.

We next turn our attention to orientations of caterpillars. A caterpillar is a tree in which every vertex is at distance at most one from a central path. Our first result is a lemma relating the dominator chromatic number of this central path to the oriented caterpillar itself.

Lemma 12. Let \(T \) be an orientation of a caterpillar and let \(P\subseteq T \) be a longest path in \(T \). Then \(\chi_{d}(T)\geq\chi_{d}(P) \).

Proof. In order for \(\chi_{d}(T) \) to be less than \(\chi_{d}(P) \) the addition of a vertex to \(P \) must allow for the combination of existing color classes in \(P \), but if this is possible then it must be possible without the addition of a new vertex which contradicts \(P \) using fewest possible colors.

Next we bound the dominator chromatic number of an oriented caterpillar by a function of the length of its central (longest) path.

Lemma 13. Let \(T \) be an orientation of a caterpillar and let \(P\subseteq T \) be its central path with length \(m \). Then \(\chi_{d}(T)\leq 2m-1 \).

Proof. Clearly \(P \) requires no more than \(m \) colors. For all vertices \(v\in V(T)\setminus V(P) \) we have that either \(d^{-}(v)=0 \) in which case all such vertices \(v \) may be assigned a new color, \(c_{m+1} \). Since any other vertices remaining in \(V(T)\setminus V(P) \) have \(d^{-}(v)=1 \), it follows that for each \(u\in V(P) \) we may color all members of \(N^{+}(u)\cap [V(T)\setminus V(P)] \) with the same color class. Since the end vertices of \(P \) do not have any such neighbors (else they are not end vertices), we may color \(V(T)\setminus V(P) \) with at most \(m-2 \) additional colors. Collectively, this worst case coloring requires \(2m-1 \) colors, thus establishing the upper bound on the dominator chromatic number of oriented caterpillars.

An interesting case of this result arises when the central path of the caterpillar is a directed path. In this case we get the rather nice result that the lower bound from Lemma 12 is sharp.

Lemma 14. If \(T \) is an orientation of a caterpillar whose central path \(P \) is a directed path, then \(\chi_{d}(T)=\chi_{d}(P) \).

Proof. Let \(P=v_{1}\dots v_{m} \). Since \(P \) is a directed path, every vertex in \(P \), except for \(v_{m} \), dominates not only a color class, but a single vertex. Notice also that the vertex \(v_{1} \) is not dominated by any other vertex, else \(P \) is not maximum (recall that \(P \) does not need to be a directed path in order to be the central path of a caterpillar). Thus every vertex in \(V(T)\setminus V(P) \) that dominates a vertex in \(P \) also dominates a color class, and may be colored with \(c(v_{1}) \). Since every vertex in \(T \) that has positive out-degree now dominates a color class, we may color all remaining vertices with \(\hat{c} \), establishing that \(\chi_{d}(T)=\chi_{d}(P)=m \).

5. Conclusion

This paper initiated the study of dominator coloring of orientations of trees. Initially, a study of arborescences and anti-arborescences found that, for a given vertex set, the dominator chromatic number of an arborescence is equal to the dominator chromatic number of an anti-arborescence. We then generalized this finding, proving the most important result in this paper, that the dominator chromatic number of orientations of trees is invariant under reversal of orientation. Using this result, several results on the dominator chromatic number of generalized stars and caterpillars were obtained.

In our study of generalized stars \(GS_{m}^{k} \), we established an upper bound on the dominator chromatic number over all orientations of \(GS_{m}^{k} \). We conjecture that this result is the best possible.

Conjecture 1. The minimum dominator chromatic number over all orientations of the generalized star \(GS_{m}^{k} \) is given by \(\chi_{d}(GS_{m}^{k})=3+m(\lfloor\frac{k}{2}\rfloor-1) \).

The study of dominator chromatic numbers of other structures which are derived from stars and generalized stars, such as wheels, are a natural extension of the results obtained on generalized stars in this paper. Additionally, the orientations of stars which are (anti-)arborescences are interesting in that they have relatively large dominator chromatic numbers. We conjecture that these two particular orientations are actually the worst possible for generalized star.

Conjecture 2. The maximum dominator chromatic number over all orientations of the generalized star \(GS_{m}^{k} \) is given by \(m(k-1)+2 \) and occurs when \(GS_{m}^{k} \) is an (anti-)arborescence.

This conjecture could prove quite insightful in determining the nature of dominating sets in (anti-)arborescences. It seems to the author that these classes of orientations of trees might be among the worst possible when it comes to finding relatively small dominating sets.

With respect to caterpillars, we established several bound on the dominator chromatic number with respect to the central path of the caterpillar. Possible directions to extend the results in this paper could include finding an explicit minimum dominator chromatic number over all orientations of caterpillars and finding explicit dominator chromatic numbers for specific caterpillars with respect to their central paths. Additionally, the study of dominator chromatic numbers of lobsters could provide a useful stepping stone from paths and caterpillars to the study of dominator chromatic numbers of orientations of trees in general, and possibly even more broadly to orientations of directed acyclic graphs.

Conflicts of Interests

The author declares no conflict of interest.

1. Introduction

Since, the foundation paper titled, Degree Tolerant Coloring of Graphs is under review, sufficient results and concepts will be recalled to makes this paper digestible. It is assumed that the reader is familiar with the definition and properties of finite Albenian groups over the operation addition \(+\), (or, multiplication \(\times\)). For ease of reference we recall that, depending on the characteristics of the elements of a non-empty set \(X\), the operations \(+\) or \(\times\) can be abstract operations. Having said that, we recall that if \(X\) is a non-empty set and \(f\) is a binary operation on \(X\) i.e., \(f:X\times X\rightarrow X\) we can denote, \(f((a,b))=a\circ b\). The ordered pair \((X,f)\) is a group if:

• (i) \(f\) is associative i.e. \(a\circ(b\circ c)=(a\circ b)\circ c\), \(\forall a,b,c\in X\).
• (ii) A \(e\in X\) exists such that, \(a\circ e=a\), \(\forall a\in X\) (called a right identity of \((X,f)\)).
• (iii) A \(b \in X\) exists for each \(a\in X\) such that, \(a\circ b=e\) (called a right inverse of \(a\)).

If for all \(a,b \in X\) we have \(a\circ b=b\circ a\) then \((X,f)\) is an Albenian group (also called a commutative group). A good introduction is found amongst others, in [1]. In this paper we will consider only a specific finite Albenian group.

It is also assumed that the reader is familiar with most of the classical concepts in graph theory. Throughout only finite, connected simple graphs will be considered. For more on general notation and concepts in graphs see [2,3,4]. It is also assumed that the reader is familiar with the concept of graph coloring. In a proper coloring of \(G\) all edges are said to be good i.e. \(\forall~uv \in E(G)\), \(c(u)\neq c(v)\). The set of colors assigned in a graph coloring is denoted by \(\mathcal{C}\) and a subset of colors assigned to a subset of vertices \(X\subseteq V(G)\) is denoted by \(c(X)\). In an improper (or defect) coloring it is permitted that for some \(uv \in E(G)\), the coloring is \(c(u)=c(v)\).

Recall that for a degree tolerant coloring abbreviated as, \(DT\)-coloring of a graph \(G\) the following condition is set:

• (i) If \(uv \in E(G)\) and \(deg(u)=deg(v)\) then, \(c(u)=c(v)\) else, \(c(u)\neq c(v)\).

Alternative formulation for condition (i). If \(uv \in E(G)\) then, \(c(u)=c(v)\) if and only if \(deg(u)=deg(v)\). The minimum number of colors which yields a \(DT\)-coloring is called the degree tolerant chromatic number of \(G\) and is denoted by, \(\chi_{dt}(G)\). A salient condition which is implied is, if \(uv \notin E(G)\) then, either \(c(u)=c(v)\) or \(c(u)\neq c(v)\). For \(K_n\), \(n\geq 1\) it easily follows that, \(\chi_{dt}(K_n) =1 \leq \chi(K_n)\). In fact, for \(n\geq 2\) it follows that, \(\chi_{dt}(K_n) < \chi(K_n)\). Furthermore, for paths \(P_1\), \(P_2\) we find \(\chi_{dt}(P_1) = \chi_{dt}(P_2) =1\). On the other hand for \(n \geq 3\) we have, \(\chi_{dt}(P_n) = \chi(P_n) = 2\). In the aforesaid it is only the coloring assignment which differs.

Consider the set \(R_u= \{deg(v): v\in N[u], u\in V(G)\}\). The degree tolerant index of \(u \in V(G)\) is defined by \(\rho(u) = |R_u|\). Note that since repetition in a set is not permitted, \(\rho(u) \leq |N[u]|\). Let the degree tolerant index of a graph \(G\) be \(\rho(G) = \max\{\rho(u):\) over all \(u\in V(G)\}\). We recall a result of interest.

Theorem 1. For a graph \(G\), it follows that \(\chi_{dt}(G)\leq \rho(G)\).

2. Albenian group \(\mathbb{Z}_n\) under addition modulo \(n\): \(n\in \mathbb{N}\)

Denote the finite Albenian group \(\mathbb{Z}_n\) under addition modulo \(n\), \(n\in \mathbb{N}\) by, \((\mathbb{Z}_{n},+_{n})\). We have \(\mathbb{Z}_n=\{0,1,2,\dots,n-1\}\) and the generator set, \(\mathbb{Z}_{gen}=\{m:m\) relatively prime to \(n\}\). Let \(\mathbb{P} =\{\)prime numbers\(\}\).

Since it is known that that any non-prime positive integer \(n\geq 4\) can be written as a product of prime numbers, begin by considering a product of two distinct primes i.e. \(n=p_1p_2\), \(p_1,p_2 \in \mathbb{P}\). It is agreed that the terms prime and prime number may be used interchangeably.

Lemma 1. For distinct positive integers \(m_1,m_2\), there are exactly \(m_1-1\) multiples of \(m_2\) and exactly \(m_2-1\) multiples of \(m_1\) in the integer range \([0,m_1m_2-1]\).

Proof. The number of multiples of \(m_1\) and \(m_2\) in the integer range \([0,m_1m_2-1]\) is given by \(\lfloor \frac{m_1p_m-1}{p_1}\rfloor\) and \(\lfloor \frac{m_1m_2-1}{m_2}\rfloor\) respectively. Further through simplification $$\lfloor \frac{m_1m_2-1}{m_1}\rfloor =m_2-1$$ and $$\lfloor \frac{m_1m_2-1}{m_2}\rfloor = m_1-1$$.

The respective sets of multiples of the primes \(p_1\) and \(p_2\) in the integer range \([0,p_1p_2-1]\) are denoted by \(M_{p_1}\) and \(M_{p_2}\). The vertex set of the power graph denoted by \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) is \(V(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = \{v_0,v_1,v_2,\dots,v_{p_1p_2-1}\}\). Alternatively, \(V_{\mathcal{P}}(\mathbb{Z}_n)\) (for brevity). The power graph has edge set \(E(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = \{v_iv_j: i\in \mathbb{Z}_{gen}\cup \{0\}, v_j \in V_{\mathcal{P}}(\mathbb{Z}_n), i\neq j\} \cup \{v_kv_\ell: k,\ell \in M_{p_1}, k \neq \ell\} \cup \{v_mv_\tau: m,\tau \in M_{p_2}, m \neq \tau\} = E_{\mathcal{P}}(\mathbb{Z}_n)\), (for brevity).

Remark 1. Conventionally the power graph is a directed graph. We only consider the undirected case, hence the underlying power graph.

Note that a result stemming from the Euler \(\varphi\)-function (see Theorem 2.13.5, [1]) yields \(|\mathbb{Z}_{gen}|=(p_1-1)(p_2-1)\). Since \(|V(\mathcal{P}((\mathbb{Z}_{n},+_{n})))| = p_1p_2\), the above implies that the vertex set of the power graph \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) can be partitioned into:

\(|V_1|= (p_2-1)\)-set and \(deg (v_i)=p_1(p_2-1)\), \(v_i\in V_1\),

\(|V_2|= (p_1-1)\)-set and \(deg(v_j)= p_2(p_1-1)\), \(v_j\in V_2\),

\(|V_{gen}\cup \{v_0\}| = (p_1p_2 -p_1-p_2+2)\)-set and \(deg(v_k)= p_1p_2-1\), \(v_k \in V_{gen}\cup \{v_0\}\).

The Figure 1 depicts \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\), \(p_1=2\), \(p_2=3\).

2.1. Degree tolerant chromatic number of \((\mathbb{Z}_{n},+_{n})\)

Lemma 2. For a positive integer \(n=p_1p_2\), \(p_1,p_2 \in \mathbb{P}\), \(p_1\neq p_2\), we have that \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = 2\).

Proof. Since \(\rho(v_i)=2\), \(v_i\in V_1\), \(\rho(v_j)=2\), \(v_j \in V_2\) and \(\rho(v_k)=3\), \(v_k \in V_{gen}\cup \{v_0\}\), it follows that \(\rho(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = 3\). From Theorem 1, \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) \leq 3\). Since \(V_1\cap V_2=\emptyset\) the coloring \(c(V_1)=c(V_2)\) is permissible in a minimal \(DT\)-coloring. Hence \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) \leq 2\). However \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) is not regular, so \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) \geq 2\). This settles the result.

Now consider \(n=p_1p_2p_3\), \(p_1,p_2,p_3 \in \mathbb{P}\) and \(p_i\neq p_j\) for all pairs.

Lemma 3. In the integer range \([0,p_1p_2p_3-1]\), let \(V_1 = \{\)multiples of \(p_1\}\), \(V_2 = \{\)multiples of \(p_2\}\), \(V_3= \{\)multiples of \(p_3\}\). Then \(|V_1 \cap V_2|=p_3-1\), \(|V_1\cap V_3|=p_2-1\) and \(|V_2\cap V_3|=p_1-1\).

Proof. Noting that without loss of generality \(V_i\cap V_j=\lfloor \frac{1}{p_i}\cdot\lfloor \frac{p_ip_jp_k -1}{p_j}\rfloor \rfloor = p_k-1\), the result follows.

Lemma 4. For a positive integer \(n=p_1p_2p_3\) and \(p_1,p_2,p_3 \in \mathbb{P}\), we have \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = 3\).

Proof. Similar to the vertex partition for the case \(n=p_1p_2\), the vertex set \(V(\mathcal{P}((\mathbb{Z}_{n},+_{n})))\) for the case \(n=p_1p_2p_3\) can be partitioned into \(V_1\), \(V_2\),\(V_3\) and \(V_{gen}\cup \{v_0\}\). Clearly, by similar reasoning found in the proof of Lemma 3, \(\chi_{dt}((\mathbb{Z}_{n},+_{n})) \leq 4\). Since no multiple of \(p_1p_2p_3\) exists, it follows that no induced \(K_3\) exists on a triple \(v_i,v_j,v_k\), \(v_i \in V_1\), \(v_j\in V_2\), \(v_k\in V_3\). Therefore \(\chi_{dt}((\mathbb{Z}_{n},+_{n})) \leq 3\).

Since \(p_i,p_j,p_k\geq 2\), it follows that \(V_i \cap V_j \neq \emptyset\). Therefore, an induced \(K_3\) exists in \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) such that each vertex has degree distinct from the other. Note that one of the vertices will be in \(V_{gen}\cup \{v_0\}\). By condition (ii), \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n})))\geq 3\). Hence the result.

We are ready to present the main result.

Theorem 2. For a positive integer \(n= \prod\limits_{i=1}^{k}p_i\), \(p_i\in \mathbb{P}\) and \(p_i \neq p_j\) iff \(i\neq j\), we have \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = k\).

Proof. The result follows by Lemmas 1,2,3,4 and the utilization of induction.

3. Conclusion

It is known that for some graphs \(\chi_{dt}(G)< \chi(G)\).

Problem 1. Show that for a positive integer \(n= \prod\limits_{i=1}^{k}p_i\), \(p_i\in \mathbb{P}\) and \(p_i \neq p_j\) iff \(i\neq j\) that \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) < \chi(\mathcal{P}((\mathbb{Z}_{n},+_{n})))\).

The numerous finite Albenian groups offer a wide scope for further research.

acknowledgments

The author would like to thank the anonymous referees for their constructive comments, which helped to improve on the elegance of this short paper.

Conflict of Interests

The author declares no conflict of interest.