Super $(a,d)$-$C_3$-antimagicness of a Corona Graph

1. Introduction

Let $G$ be a simple graph with vertex set $V$ and edge set $E$. An edge-covering of finite and simple graph $G$ is a family of subgraphs $H_1, H_2, \dots,H_t$ such that each edge of $E(G)$ belongs to at least one of the subgraphs $H_i$, $i=1, 2, \dots, t$. In this case we say that $G$ admits an $(H_1, H_2, \dots, H_t)$-(edge) covering. If every subgraph $H_i$ is isomorphic to a given graph $H$, then the graph $G$ admits an $H$-covering. A graph $G$ admitting an $H$-covering is called $(a,d)$-$H$-antimagic if there exists a total labeling $f:V(G)\cup E(G) \to \{1,2,\dots, |V(G)|+|E(G)| \}$ such that for each subgraph $H'$ of $G$ isomorphic to $H$, the $H'$-weights, $$wt_f(H')= \sum\limits_{v\in V(H')} f(v) + \sum\limits_{e\in E(H')} f(e),$$ constitute an~arithmetic progression $a, a+d, a+2d,\dots , a+(t -1)d$, where $a>0$ and $d\ge 0$ are two integers and $t$ is the number of all subgraphs of $G$ isomorphic to $H$.

The (super) $H$-magic graph was first introduced by Gutiérrez and Lladó in [1]. The $(a,d)$-$H$-antimagic labeling was introduced by Inayah et al. [2].

In [3] Bača et al. investigated the super tree-antimagic total labelings of disjoint union of graphs. Bača et al. [4] showed the constructions for $H$-antimagicness of Cartesian product of graphs. In [5], authors proved the $C_n$-antimagicness of Fan graph for several difference depending on the length of the cycle. In [6, 7, 8] Umar et al. proved the existence of super $(a,1)$-Tree-antimagicness of Sun graphs, super $(a,d)$-$C_n$-antimagicness of Windmill graphs for several differences and super $(a,d)$-$C_4$-antimagicness of Book graph and their disjoint union.

In this paper, we study the existence of super $(a,d)$-$C_3$-antimagic labeling of a special type of a corona graph.

2. Super Cycle-antimagic labeling of Corona graph

The join of two graphs $H_1$ and $H_2$, denoted by $H_1+H_2$, is the graph where $V(H_1) \cap V(H_2)= \emptyset$ and each vertex of $H_1$ is adjacent to all vertices of $H_2$ [9]. When $H_1=K_1$, this is the corona graph $K_1 \odot H_2$. In this paper, we consider a special type of a corona graph.

Let $K_1$ be a complete graph and $S_n$ be a star on $n+1$ vertices. We consider the corona graph $G= K_1 \odot S_n$, where $$V(G):=\{v_1,v_2,x_1,x_2,\dots,x_n\}$$ and $$E(G):=\{v_1v_2,v_1x_1,v_1x_2,\dots,v_1x_n,v_2x_1,v_2x_2,\dots,v_2x_n\}$$ The corona graph $G$ is covered by the cycles $C_3^{(i)}$, $1\leq i \leq n$ and the $C_3^{(i)}$-weights under a labeling $h$ is:

2.1. $C_3$-Supermagic labeling

Theorem 2.1. Let $G:=K_1\odot S_n$ be a corona graph of $K_1$ and $S_n$ and $n \geq 2$ be an integer then the graph $G$ admits a $C_3$-supermagic labeing.

Proof. $n \equiv 0 (\text{mod}\ 2)$
The labeling $h_0$ is defined as: $$\begin{align*} h_0(v_1)&=1,\\ h_0(v_2)&=\frac{n}{2}+2,\\ h_0(v_1v_2)&= 3n+3,\\ h_0(v_1x_i)&=3n+3-i.\\ \end{align*}$$ $$h_0(x_{i})= \begin{cases} \frac{n}{2}+2-i \ \ & \ \ \ \ \ \textrm{ if $i = 1,2,\dots, \frac{n}{2}$} \\ \frac{3n+6}{2}-i \ \ & \ \ \ \ \ \textrm{ if $i = \frac{n}{2}+1,\frac{n}{2}+2, \dots, n$} \\ \end{cases}$$ $$h_0(v_2x_{i})= \begin{cases} n+2(1+i) \ \ & \textrm{ if $i = 1,2,\dots, \frac{n}{2}$} \\ 2i+1 \ \ & \textrm{ if $i = \frac{n}{2}+1,\frac{n}{2}+2, \dots, n$} \\ \end{cases}$$ Clearly, the vertices assume least possible integers $\{1,2,\dots, n+2\}$ under the labeling $h_0$ and edges receive the labels $\{n+3, n+4,\dots, 3n+3\}$. Therefore $h_0$ is a super total labeling.
Using equation (1)

When $n \equiv 1 \ \ (\text{mod}\ 2)$
The labeling $h_0$ is defined as: $$\begin{align*} h_0(v_i)&=i,\\ h_0(v_1v_2)&= n+3,\\ h_0(x_i)&=n+3-i.\\ \end{align*}$$ For $i \equiv 0$ (mod $2$) $$h_0(v_jx_{i})= \begin{cases} n+3 +\frac{i}{2} \ \ & \textrm{ if $j = 1$} \\ \frac{5n+7+i}{2} \ \ & \textrm{ if $j = 2$} \\ \end{cases}$$ For $i \equiv 1$ (mod $2$) $$h_0(v_jx_{i})= \begin{cases} \frac{3(n+2)+i}{2} \ \ & \textrm{ if $j = 1$} \\ \frac{4n+7+i}{2} \ \ & \textrm{ if $j = 2$} \\ \end{cases}$$ Clearly, the vertices assume least possible integers $\{1,2,\dots, n+2\}$ under the labeling $h_0$ and edges receive the labels $\{n+3, n+4,\dots, 3n+3\}$. Therefore $h_0$ is a super total labeling.
Using equation (1) Equations (2, 3) shows $wt_{h_0}(C_3^{(i)})$ is independent of $i$. Hence the corona graph $G$ admits a $C_3$-supermagic labeling. This completes the proof.

2.2. Super $(a, d)$-$C_3$-antimagic labeling

Theorem 2.2. Let $G:=K_1\odot S_n$ be a corona graph of $K_1$ and $S_n$ and $n \geq 2$ be an integer then the graph $G$ admits a super $(a,1)$-$C_3$-antimagic labeing.

Proof. The labeling $h_1$ is defined as: $$\begin{align*} h_1(v_i)&=i,\\ h_1(v_1v_2)&= n+3,\\ h_1(v_2x_{i})&= 2n+3+i. \end{align*}$$ $$h_1(x_{i})= \begin{cases} \frac{i+1}{2}+2 \ \ & \textrm{ if $i \equiv 1$ (mod \ $2$)} \\ \lceil\frac{n}{2}\rceil+ 2 +\frac{i}{2} \ \ & \textrm{ if $i \equiv 0$ (mod $2$)} \\ \end{cases}$$ $$h_1(v_1x_{i})= \begin{cases} \frac{4n+7-i}{2} \ \ & \textrm{ if $i \equiv 1$ (mod \ $2$)} \\ \lceil\frac{n-1}{2}\rceil+ n+4-\frac{i}{2} \ \ & \textrm{ if $i \equiv 0$ (mod $2$)} \\ \end{cases}$$ Clearly, the vertices assume least possible integers $\{1,2,\dots, n+2\}$ under the labeling $h_1$ and edges receive labels $\{n+3, n+4, \dots, 3n+3\}$. Therefore $h_1$ is a super total labeling.
Using equation (1)

Equation (4) shows $wt_{h_0}(C_3^{(i)})$ constitute an arithmetic progression with $a=5(n+3)+1$ and $d=1$. Hence the corona graph $G$ admits a super $(a,1)$-$C_3$-antimagic labeling. This completes the proof.

Theorem 2.3. Let $G:=K_1\odot S_n$ be a corona graph of $K_1$ and $S_n$ and $n \geq 2$ be an integer then the graph $G$ admits a super $(a,d)$-$C_3$-antimagic labeing for $d=3,5$.

Proof. The labeling $h_d$ is defined as: $$\begin{align*} h_d(v_i)&=i,\\ h_d(v_1v_2)&= n+3,\\ h_d(x_i)&= 2+i. \end{align*}$$ $$h_3(v_jx_{i})= \begin{cases} 2n+3+i \ \ & \textrm{ if $j=1$} \\ n+3+i \ \ & \textrm{ if $j=2$} \\ \end{cases}$$ $$h_5(v_jx_{i})= \begin{cases} n+2+2i \ \ & \textrm{ if $j=1$} \\ n+3+2i \ \ & \textrm{ if $j=2$} \\ \end{cases}$$ Clearly, the vertices assume least possible integers $\{1,2,\dots, n+2\}$ under the labeling $h_d$ and edges receive labels $\{n+3, n+4, \dots, 3n+3\}$. Therefore $h_d$ is a super total labeling.
Using equation (1)

Equation (5) shows $wt_{h_3}(C_3^{(i)})$ constitute an arithmetic progression with $a=2(2n+7)+3$ and $d=3$. Hence the corona graph $G$ admits a super $(a,3)$-$C_3$-antimagic labeling.
Now, for case $d=5$, Using equation (1) Equation (6) shows $wt_{h_3}(C_3^{(i)})$ constitute an arithmetic progression with $a=3(n+6)$ and $d=5$. Hence the corona graph $G$ admits a super $(a,5)$-$C_3$-antimagic labeling. This completes the proof.

Theorem 2.4. Let $G:=K_1\odot S_n$ be a corona graph of $K_1$ and $S_n$ and $n \geq 2$ be an integer then the graph $G$ admits a super $(a,d)$-$C_3$-antimagic labeing for $d=2,4$.

Proof. The labeling $h_d$ is defined as: $$h_d(v_i)=i$$ $$h_d(x_{i})= \begin{cases} n+3-i \ \ & \textrm{ if $d=2$} \\ 2+i \ \ & \textrm{ if $d=4$} \\ \end{cases}$$ The edges are labeled as:
When $n \equiv 0 \ \ \ (\text{mod} \;2)$
$$h_d(v_1v_2)= 5\left(\frac{n}{2}\right)+3$$ $$h_d(v_1x_{i})= \begin{cases} n + 2 + i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n}{2}+1$ }\\ \frac{n}{2}+1+2i \ \ & \textrm{ if $i = \frac{n}{2} +2, \frac{n}{2} +3,...,n$}\\ \end{cases}$$ $$h_d(v_2x_{i})= \begin{cases} \frac{3n}{2}+ 2(1 + i) \ \ & \textrm{ if $i = 1,2,\dots,\frac{n}{2}+1$ }\\ 2n+3+i \ \ & \textrm{ if $i = \frac{n}{2}+2, \frac{n}{2}+3,...,n$}\\ \end{cases}$$ Clearly, the vertices assume least possible integers $\{1,2,\dots, n+2\}$ under the labeling $h_d$ and edges receive labels $\{n+3, n+4, \dots, 3n+3\}$. Therefore $h_d$ is a super total labeling.
Using equation (1)

Equation (7) shows $wt_{h_2}(C_3^{(i)})$ constitute an arithmetic progression with $a=6n+15$ and $d=2$. Hence the corona graph $G$ admits a super $(a,2)$-$C_3$-antimagic labeling. Using equation (1) Equation (8) shows $wt_{h_4}(C_3^{(i)})$ constitute an arithmetic progression with $a=5n+16$ and $d=4$. Hence the corona graph $G$ admits a super $(a,4)$-$C_3$-antimagic labeling.
When $n \equiv 1 \ \ (\text{mod}\; 2)$
$$h_d(v_1v_2)= 3n+3$$ $$h_d(v_1x_{i})= \begin{cases} n + 2 + i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n+1}{2}$}\\ \frac{n+1}{2} + 1 + 2i \ \ & \textrm{ if $i = \frac{n+1}{2}+1, \frac{n+1}{2} +2,...,n$}\\ \end{cases}$$ $$h_d(v_2x_{i})= \begin{cases} \frac{n+1}{2}+n+1+2i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n+1}{2}$}\\ 2(n+1)+i \ \ & \textrm{ if $i = \frac{n+1}{2}+1, \frac{n+1}{2}+2,...,n$}\\ \end{cases}$$ Clearly, the vertices assume least possible integers $\{1,2,\dots, n+2\}$ under the labeling $h_d$ and edges receive labels $\{n+3, n+4, \dots, 3n+3\}$. Therefore $h_d$ is a super total labeling.
Using equation (1) Equation (9) shows $wt_{h_2}(C_3^{(i)})$ constitute an arithmetic progression with $a=\frac{13n+29}{2}$ and $d=2$. Hence the corona graph $G$ admits a super $(a,2)$-$C_3$-antimagic labeling.
Using equation (1) Equation (10) shows $wt_{h_4}(C_3^{(i)})$ constitute an arithmetic progression with $a=\frac{11n+31}{2}$ and $d=4$. Hence the corona graph $G$ admits a super $(a,4)$-$C_3$-antimagic labeling. This completes the proof.

Competing Interests

The author(s) do not have any competing interests in the manuscript.