1. Introduction

Machine learning is a field of artificial intelligence which gives computer systems the ability to ''learn'' using available data. Recently machine learning algorithms become very popular for analyzing of data and make prediction [1,2,3,4]. Linear models for classification is a part of supervised learning [1]. Supervised learning is machine learning task of learning a function which transforms an input to an output data using available input-output data. In supervised learning, every example is a pair consisting of an input object (typically a vector) and a desired output value (also called the supervisory signal). A supervised learning algorithm analyzes the training data and produces an inferred function, which can be used then for analyzing of new examples. Supervised Machine Learning algorithms include linear and logistic regression, multi-class classification, decision trees and support vector machines. In this work we will concentrate attention on study the linear and logistic regression algorithms. Supervised learning problems are further divided into Regression and Classification problems. Both problems have as goal the construction of a model which can predict the value of the dependent attribute from the attribute variables. The difference between these two problems is the fact that the attribute is numerical for regression and logical (belonging to class or not) for classification.

In this work are studied linear and polynomial classifiers, more precisely, the regularized versions of least squares and perceptron learning algorithms. The WINNOW algorithm for classification is also presented since it is used in numerical examples of Section 6 for comparison of different classification strategies. The classification problem is formulated as a regularized minimization problem for finding optimal weights in the model function. To formulate iterative gradient-based classification algorithms the Fréchet derivatives for the non-regularized and regularized least squares algorithms are presented. The Fréchet derivative for the perceptron algorithm is also rigorously derived.

Adding the regularization term in the functional leads to the optimal choice of weights such that they make a trade-off between observed data and obtaining a minimum of this functional. Different rules are used for choosing the regularization parameter in machine learning, and most popular are early stopping algorithm, bagging and dropout techniques [5], genetic algorithms [6], particle swarm optimization [7,8], racing algorithms [9] and Bayesian optimization techniques [10,11]. In this work are presented the most popular a priori and a posteriori Tikhonov's regularization rules for choosing the regularization parameter in the cost functional. Finally, performance of non-regularized versions of all classification algorithms with respect to applicability, reliability and efficiency is analyzed on simulated and experimental data sets [12,13].

The outline of the paper is as follows. In Section 2 are briefly formulated non-regularized and regularized classification problems. Least squares for classification are discussed in Section 3. Machine learning linear and polynomial classifiers are presented in Section 4. Tikhonov's methods of regularization for classification problems are discussed in Section 5. Finally, numerical tests are presented in Section 6.

2. Classification problem

The goal of regression is to predict the value of one or more continuous target variables \(t=\{t_i\}, i=1,...,m\) by knowing the values of input vector \(x=\{x_i\}, i=1,...,m\). Usually, classification algorithms are working well for linearly separable data sets.

Definition 1. Let \(A\) and \(B\) are two data sets of points in an \(n\)-dimensional Euclidean space. Then \(A\) and \(B\) are linearly separable if there exist \(n + 1\) real numbers \(\omega_1, ..., \omega_n, l\) such that every point \(x \in A\) satisfies \(\sum _{i=1}^{n}\omega_{i} x_{i} > l\) and every point \(x \in B\) satisfies \(\sum _{i=1}^{n}\omega_{i} x_{i} < -l\).

The classification problem is formulated as follows:

• Suppose that we have data points \(\{x_i\}, i=1,...,m\) which are separated into two classes \(A\) and \(B\). Assume that these classes are linearly separable.
• Our goal is to find the decision line which will separate these two classes. This line will also predict in which class will the new point fall.

In the non-regularized classification problem the goal is to find optimal weights \(\omega=(\omega_1,...,\omega_M)\), \(M\) is the number of weights, in the functional

with \(m\) data points. Here, \( t =\{t_i\}, i = 1,...,m\), is the target function with known values, \( y(\omega) = \{ y_i(\omega)\} := \{ y(x_i, \omega)\}, i = 1,...,m\), is the classifiers model function.

To find optimal vector of weights \(\omega = \{\omega_i\}, i=1,...,M\) in the regularized classification problem, the regularization term is added to the functional (1):

Here, \(\gamma\) is the regularization parameter, \(\| \omega \|^2 = \omega^T \omega = \omega_1^2 + ... + \omega_M^2\), \(M\) is the number of weights. In order to find the optimal weights in (1) or in (2), the following minimization problem should be solved

Thus, we seek for a stationary point of (1) or (4) with respect to \(\omega\) such that

where \(F^{\prime }(\omega)\) is the Fréchet derivative acting on \(\bar{\omega}\).

More precisely, for the functional (4) we get

The Fréchet derivative of the functional (1) is obtained by taking \(\gamma=0\) in (7). To find optimal vector of weights \(\omega = \{\omega_i\}, i=1,...,M\) can be used the Algorithm 1 as well as least squares or machine learning algorithms.

For computation of the learning rate \(\eta\) in the Algorithm 1 usually is used optimal rule which can be derived similarly as in [14]. However, as a rule take \(\eta=0.5\) in machine learning classification algorithms [4]. Among all other regularization methods applied in machine learning [5,6,7,8,9,10,11], the regularization parameter \(\gamma\) can be also computed using the Tikhonov's theory for inverse and ill-posed problems by different algorithms presented in [15,16,17,18,19]. Some of these algorithms are discussed in Section 5.

3. Least squares for classification

The linear regression is similar to the solution of linear least squares problem and can be used for classification problems appearing in machine learning algorithms. We will revise solution of linear least squares problem in terms of linear regression.

The simplest linear model for regression is

Here, \(\omega = \{ \omega_i \}, i=0,..., M\) are weights with bias parameter \(\omega_0\), \(\{x_i\}, i=1,..., M\) are training examples. Target values (known data) are \(\{t_i\}, i=1,...,N\) which correspond to \(\{x_i\}, i=1,...,M\). Here, \(M\) is the number of weights and \(N\) is the number of data points. The goal is to predict the value of \(t\) in (1) for a new value of \(x\) in the model function (8).

The linear model (8) can be written in the form

where \(\varphi_i(x), i=0,...,M\) are known basis functions with \(\varphi_0(x) = 1\).

3.1. Non-regularized least squares problem

In non-regularized linear regression or least squares problem the goal is to minimize the sum of squares

to find optimal weights \(\omega_i, i= 0,..., M\), in the minimization problem for points \(x_n, n=1,...,N\). The problem (11) is a typical least squares problem of the minimizing the squared residuals with the residual \(r(\omega) = t - \omega^T \varphi(x)\). The test functions \(\varphi(x)\) form the design matrix \(A\) and the regression problem (or the least squares problem) is written as where \(A\) is of the size \(N \times M\) with \(N > M\), \(t\) is the target vector of the size \(N\), and \(\omega\) is vector of weights of the size \(M\).

To find minimum of the error function (10) and derive the normal equations, we look for the \(\omega\) where the gradient of the norm \(\| r(\omega)\|_2^2 = ||A \omega - t||^2_2 = (A\omega - t)^T(A\omega - t)\) vanishes, or where \((\|r(\omega) \|_2^2)'_\omega =0\). To derive the Fréchet derivative, we consider the difference \(\| r(\omega + e)\|_2^2 - \| r(\omega)\|_2^2\) and single out the linear part with respect to \(\omega\). More precisely, we get

\begin{equation*} \begin{split} 0 = &{\displaystyle\lim_{\|e\| \rightarrow 0}}\dfrac{(A(\omega+e)- t)^T(A(\omega + e) - t)-(A\omega- t)^T(A\omega - t)}{||e||_2} \\ = &{\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{((A \omega - t) + Ae )^T((A \omega - t) + Ae)-(A\omega- t)^T(A\omega - t)}{||e||_2} \\ =& {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{\|(A \omega - t) + Ae \|_2^2 - \| A\omega - t \|_2^2}{||e||_2} \\ &= {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{\|A \omega - t\|_2^2 + 2 (A \omega - t ) \cdot Ae + \| Ae \|_2^2 - \| A\omega - t \|_2^2}{||e||_2} \\ = & {\displaystyle\lim_{\|e\| \rightarrow 0}}\dfrac{2e^T(A^TA \omega - A^T t)+e^TA^TAe}{||e||_2} \end{split} \end{equation*} Thus, The second term in (15) can be estimated as Thus, the first term in (15) must also be zero, and thus,

Equations (17) is a symmetric linear system of the \(M \times M \) linear equations for \(M\) unknowns called normal equations.

Using definition of the residual in the functional can be computed the Hessian matrix \(H = A^T A\). If the Hessian matrix \(H = A^T A\) is positive definite, then \(\omega\) is indeed a minimum.

Lemma 1. The matrix \(A^T A\) is positive definite if and only if the columns of \(A\) are linearly independent, or when \(rank(A)=M\) (full rank).

Proof. We have that \(dim(A) = N \times M\), and thus, \(dim(A^T A) = M \times M\). Thus, \(\forall v \in R^M\) such that \(v \neq 0\)

For positive definite matrix \(A^TA\) we need to show that \(v^T A^T A v > 0\). Assume that \(v^T A^T A v = 0\). We observe that \(A v = 0\) only if the linear combination \(\sum_{i=1}^M a_{ji} v_i = 0\). Here, \(a_{ji}\) are elements of row \(j\) in \(A\). This will be true only if columns of \(A\) are linearly dependent or when \(v=0\), but this is contradiction with assumption \(v^T A^T A v = 0\) since \(v \neq 0\) and thus, the columns of \(A\) are linearly independent and \(v^T A^T A v > 0\).

The final conclusion is that if the matrix \(A\) has a full rank (\(rank(A)=M\)) then the system (17) is of the size \(M\)-by-\(M\) and is symmetric positive definite system of normal equations. It has the same solution \(\omega\) as the least squares problem \( \min_\omega \|A \omega - t\|^2_2\) and can be solved efficiently via Cholesky decomposition [20].

3.2. Regularized linear regression

Let now the matrix \(A\) will have entries \(a_{ij} = \phi_j(x_i), i=1,...,N; j = 1,...,M\). Recall, that functions \(\phi_j(x), j = 0,...,M\) are called basis functions which should be chosen and are known. Then the regularized least squares problem takes the form

To minimize the regularized squared residuals (20) we will again derive the normal equations. Similarly as was derived the Fréchet derivative for the non-regularized regression problem (14), we look for the \(\omega\) where the gradient of \(\frac{1}{2} ||A \omega - t||^2_2 + \frac{\gamma}{2} \| \omega \|_2^2 = \frac{1}{2} (A\omega - t)^T(A\omega - t) + \frac{\gamma}{2} \omega^T \omega\) vanishes. In other words, we consider the difference \( (\| r(\omega + e)\|_2^2 + \frac{\gamma}{2} \| \omega + e \|_2^2) - (\| r(\omega)\|_2^2 + \frac{\gamma}{2} \| \omega \|_2^2)\), then single out the linear part with respect to \(\omega\) to obtain:

\begin{equation*} \begin{split} 0 = &\frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{(A(\omega+e)- t)^T(A(\omega + e) - t)-(A\omega- t)^T(A\omega - t)}{||e||_2} + {\displaystyle \lim_{\|e\| \rightarrow 0}} \dfrac{\frac{\gamma}{2} (\omega +e)^T(\omega + e) - \frac{\gamma}{2} \omega^T \omega}{||e||_2} \\ =& \frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{\|(A \omega - t) + Ae \|_2^2 - \| A\omega - t \|_2^2}{||e||_2} + {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{ \frac{\gamma}{2}(\| \omega + e\|_2^2 - \| \omega \|_2^2)}{ \|e\|_2} \\ =& \frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{\|A \omega - t\|_2^2 + 2 (A \omega - t ) \cdot Ae + \| Ae \|_2^2 - \| A\omega - t \|_2^2}{||e||_2} + \frac{\gamma}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{\|\omega \|_2^2 + 2 e^T \omega + \|e \|_2^2 - \| \omega \|_2^2}{||e||_2} \end{split} \end{equation*} \begin{equation*} = \frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}}\dfrac{2e^T(A^TA \omega - A^T t)+e^TA^TAe}{||e||_2} + \frac{\gamma}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}}\dfrac{2e^T \omega + e^T e}{||e||_2}. \end{equation*} The term Similarly, the term We finally get \begin{equation*} \begin{split} 0 = {\displaystyle \lim_{\|e\| \rightarrow 0}} \dfrac{e^T(A^TA \omega - A^T t)}{||e||_2} + \dfrac{\gamma e^T \omega }{||e||_2}. \end{split} \end{equation*}

The expression above means that the factor \(A^TA \omega - A^T t + \gamma \omega \) must also be zero, or \( (A^T A + \gamma I) \omega = A^T t, \) where \(I\) is the identity matrix. This is a system of \(M\) linear equations for \(M\) unknowns, the normal equations for regularized least squares.

Figure 1 shows that the linear regression or least squares minimization \(\min_\omega \| A \omega - t\|_2^2\) for classification is working fine when it is known that two classes are linearly separable. Here the linear model equation in the problem (12) is

and the target values of the vector \(t=\{t_i\}, i=1,...,N\) in (12) are The elements of the design matrix (13) are given by

3.3. Polynomial fitting to data in two-class model

Let us consider the least squares classification in the two-class model in the general case. Let the first class consisting of \(l\) points with coordinates \((x_i, y_i), i=1,...,l\) is described by it's linear model Let the second class consisting of \(k\) points with coordinates \((x_i, y_i), i=1,...,k\) is also described by the same linear model Here, basis functions are \(\phi_j(x), j = 1,...,n\). Our goal is to find the vector of parameters \(c= c_{i,1} = c_{i,2}, i=1,...,n\) of the size \(n\) which will fit best to the data \(y_i, i=1,...,m, m=k+l\) of both model functions, \(f_1(x_i, c), i=1,...,l\) and \(f_2(x_i, c), i=1,...,k\) with \(f(x,c) = [f_1(x_i, c), f_2(x_i,c)]\) such that the minimization problem is solved with \(m=k+l\). If the function \(f(x, c)\) in (28) is linear then we can reformulate the minimization problem (28) as the following least squares problem where the matrix \(A\) in the linear system \begin{equation*} Ac=y \end{equation*} will have entries \(a_{ij} = \phi_j(x_i), i=1,...,m; j = 1,...,n\), i.e. elements of the matrix \(A\) are created by basis functions \(\phi_j(x), j = 1,...,n\). Solution of (29) can be found by the method of normal equations derived in Section 3.1: with pseudo-inverse matrix \(A^+ := (A^T A)^{-1} A^T\).

For creating of elements of \(A\) different basis functions can be chosen. The polynomial test functions

have been considered in the problem of fitting to a polynomial in examples presented in Figures 2. The matrix \(A\) constructed by these basis functions is a Vandermonde matrix, and problems related to this matrix are discussed in [20]. Linear splines (or hat functions) and bellsplines also can be used as basis functions [20].

Figures 2 present examples of polynomial fitting to data for two-class model with \(m=10\) using basis functions \(\phi_j(x) =x^{j-1}, j=1,...,d\), where \(d\) is degree of the polynomial. Using these figures we observe that least squares fit data well and even can separate points in two different classes, although this is not always the case. Higher degree of polynomial separates two classes better. However, since Vandermonde's matrix can be ill-conditioned for high degrees of polynomial, we should carefully choose appropriate polynomial to fit data.

4. Machine learning linear and polynomial classifiers

In this section we will present the basic machine learning algorithms for classification problems: perceptron learning and WINNOW algorithms. Let us start with considering of an example: determine the decision line for points presented in Figure 3. One example on this figure is labeled as positive class, another one as negative. In this case, two classes are separated by the linear equation with three weights \(\omega_i, i=1,2,3\), given by

In common case, two classes can be separated by the general equation which also can be written as with \(x_0=1\). If \(n=2\) then the above equation defines a line, if \(n=3\) - plane, if \(n>3\) - hyperplane. The problem is to determine weights \(\omega_i\) and the task of machine learning is to determine their appropriate values. Weights \(\omega_i, i=1,...,n\) determine the angle of the hyperplane, \(\omega_0\) is called bias and determines the offset, or the hyperplanes distance from the origin of the system of coordinates.

4.1. Perceptron learning for classification

The main idea of perceptron is binary classification. The perceptron computes a sum of weighted inputs and uses the binary classification When weights are computed, the linear classification boundary is defined by \begin{equation*} h(x,\omega) = \omega^T x = 0. \end{equation*} Thus, the perceptron algorithm determines weights in (36) via binary classification (37).

Binary classifier \(y(x,\omega)\) in (37) decides whether or not an input \(x\) belongs to some specific class:

where \(\omega_0\) is the bias. The bias does not depend on the input value \(x\) and shifts the decision boundary. If the learning sets are not linearly separated the perceptron learning algorithm does not terminate and will never converge and classify data properly, see Figure 7-a).

The algorithm which determines weights in (36) via binary classification (37) can be reasoned by minimization of the regularized residual

where \(\xi_\delta(x)\) for a small \(\delta\) is a data compatibility function to avoid discontinuities which can be defined similarly with [21] and \(\gamma\) is the regularization parameter. Taking \(\gamma=0\) algorithm will minimize the non-regularized residual (39). Alternative, it can be minimized the residual over the set \(M \subset \{ 1,..., m\} \) of the currently miss-classified patterns which is simplified for the case of the perceptron algorithm to the minimization of the following residual with \begin{equation*} |- t_i h_i|_+ = \max (0,- t_i h_i ), i = 1,...,m. \end{equation*} The Algorithm 2 presents the regularized perceptron learning algorithm where in update of weights (32) was used the following regularized functional Here, \(\gamma\) is the regularization parameter, \(t\) is the target function, or class \(c\) in the algorithm 2, which takes values \(0\) or \(1\).

To find optimal weights in (42) we need to solve the minimization problem in the form (5)

where \(F'(\omega)\) is a Frechet derivative acting on \(\bar{\omega}\). To derive \(F'(\omega)\) for (39) we seek the \(\omega\) where the gradient of \(\frac{1}{2} ||r(x, \omega)||^2_2 + \frac{\gamma}{2} \| \omega \|_2^2\) vanishes. In other words, we consider for (39) the difference \( (\| r(x, \omega + e)\|_2^2 + \frac{\gamma}{2} \| \omega + e \|_2^2) - (\| r(x, \omega)\|_2^2 + \frac{\gamma}{2} \| \omega \|_2^2)\), then single out the linear part with respect to \(\omega\) to obtain: \begin{equation*} \begin{split} 0 = &\frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{ \|(t - y(x,\omega + e))\xi_\delta(x)\|_2^2 + \frac{\gamma}{2} \| \omega + e \|_2^2 - \|(t - y(x,\omega))\xi_\delta(x)\|_2^2 - \frac{\gamma}{2} \| \omega \|_2^2 }{||e||_2} \\ = &\frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{ \|(t - \sum_{i=0}^n \omega_i x_i - \sum_{i=0}^n e_i x_i)\xi_\delta(x)\|_2^2 - \|(t - y(x,\omega))\xi_\delta(x)\|_2^2 }{||e||_2} + {\displaystyle \lim_{\|e\| \rightarrow 0}} \dfrac{\frac{\gamma}{2} (\omega +e)^T(\omega + e) - \frac{\gamma}{2} \omega^T \omega}{||e||_2} \\ = &\frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{ \|(t - y(x,\omega) - e^T x)\xi_\delta(x)\|_2^2 - \|(t - y(x,\omega))\xi_\delta(x)\|_2^2}{||e||_2} + {\displaystyle \lim_{\|e\| \rightarrow 0}} \dfrac{\frac{\gamma}{2} (\omega +e)^T(\omega + e) - \frac{\gamma}{2} \omega^T \omega}{||e||_2} \\ =& \frac{1}{2} {\displaystyle \lim_{\|e\| \rightarrow 0}} \dfrac{\|(t - y(x,\omega))~\xi_\delta(x)\|_2^2 - 2(t - y(x,\omega))\cdot e^Tx~\xi_\delta(x) + \| e^T x~\xi_\delta(x)\|_2^2}{||e||_2} \\ -& \frac{1}{2} {\displaystyle \lim_{\|e\| \rightarrow 0}}\dfrac{\|(t - y(x,\omega))~\xi_\delta(x)\|_2^2}{||e||_2} + {\displaystyle \lim_{\|e\| \rightarrow 0}} \dfrac{\frac{\gamma}{2} (\omega +e)^T(\omega + e) - \frac{\gamma}{2} \omega^T \omega}{||e||_2} \\ =& \frac{1}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}} \dfrac{ - 2 (t - y(x,\omega))\cdot e^Tx~ \xi_\delta(x) + \| e^T x~\xi_\delta(x)\|_2^2}{||e||_2} + \frac{\gamma}{2} {\displaystyle\lim_{\|e\| \rightarrow 0}}\dfrac{2e^T \omega + e^T e}{||e||_2}. \end{split} \end{equation*} The second part in the last term of the above expression is estimated as in (22). The second part in the first term is estimated as We finally get \begin{equation*} \begin{split} 0 = {\displaystyle \lim_{\|e\| \rightarrow 0}} -\dfrac{x^T e( t - y(x,\omega)) ~\xi_\delta(x)}{||e||_2} + \dfrac{\gamma e^T \omega }{||e||_2}. \end{split} \end{equation*} The expression above means that the factor \(- x^T ( t - y(x,\omega)) ~\xi_\delta(x) + \gamma \omega \) must also be zero, or The non-regularized version of the perceptron Algorithm 2 is obtained taking \(\gamma=0\) in (45).

4.2. Polynomial of the second order

Coefficients of polynomials of the second order can be obtained by the same technique as coefficients for linear classifiers. The second order polynomial function is:

This polynomial can be converted to the linear classifier if we introduce notations: \[ z_1 = x_1, z_2=x_2, z_3= x_1^2, z_4 = x_1 x_2, z_5= x_2^2. \] Then equation (46) can be written in new variables as which is already the linear function. Thus, the Perceptron learning Algorithm 2 can be used to determine weights \(\omega_0, ..., \omega_5\) in (47).

Suppose that weights \(\omega_0, ..., \omega_5\) in (47) are computed. To present the decision line one need to solve the following quadratic equation for \(x_2\):

with known weights \(\omega_0, ..., \omega_5\) and known \(x_1\) which can be rewritten as or in the form with known coefficients \( a= \omega_5, b = \omega_2 + \omega_4 x_1, c = \omega_0 + \omega_1 x_1 + \omega_3 x_1^2\). Solutions of (50) will be Thus, to present the decision line for polynomial of the second order, first should be computed weights \(\omega_0, ..., \omega_5\), and then the quadratic equation (49) should be solved the solutions of which are given by (51). Depending on the classification problem and set of admissible parameters for classes, one can then decide which one classification line should be presented, see examples in section 6.

4.3. WINNOW learning algorithm

To be able compare perceptron with other machine learning algorithms, we present here one more learning algorithm which is very close to the perceptron and called WINNOW. Here is described the simplest version of this algorithm without regularization. The regularized version of WINNOW is analyzed in [22]. Perceptron learning algorithm uses additive rule in the updating weights, while WINNOW algorithm uses multiplicative rule: weights are multiplied in this rule. The WINNOW algorithm Algorithm 3 is written for \(c=t\) and \(y=h\) in (45). We will again assume that all examples where \(c(\textbf{x})=1\) are linearly separable from examples where \(c(\textbf{x})=0\).

5. Methods of Tikhonov's regularization for classification problems

To solve the regularized classification problem the regularization parameter \(\gamma\) can be chosen by the same methods which are used for the solution of ill-posed problems. For different Tikhonov's regularization strategies we refer to [15,16,17,18,19,23]. In this section we will present main methods of Tikhonov's regularization which follows ideas of [15,16,17,19,23].

Definition 2. Let \(B_{1}\) and \(B_{2}\) be two Banach spaces and \(G\subset B_{1}\) be a set. Let \(y:G\rightarrow B_{2}\) be one-to-one. Consider the equation

where \(t\) is the target function and \(y(\omega)\) is the model function in the classification problem. Let \(t^{\ast }\) be the noiseless target function in equation (52) and \(\omega^{\ast }\) be the ideal noiseless weights corresponding to \( t^{\ast }, ~y(\omega^{\ast }) = t^{\ast}\). For every \(\delta \in \left( 0,\delta _{0}\right), ~\delta _{0}\in \left( 0,1\right) \) denote \begin{equation*} K_{\delta }( t^{\ast }) = \left\{ z\in B_{2}:\left\Vert z- t^{\ast }\right\Vert _{B_{2}}\leq \delta \right\}. \end{equation*} Let \(\gamma >0\) be a parameter and \(R_{\gamma}:K_{\delta _{0}}(t^{\ast}) \rightarrow G\) be a continuous operator depending on the parameter \(\gamma\). The operator \(R_{\gamma }\) is called the regularization operator for (52) if there exists a function \(\gamma_0 \left( \delta \right) \) defined for \(\delta \in \left( 0,\delta _{0}\right) \) such that \begin{equation*} \lim_{\delta \rightarrow 0} \left\Vert R_{\gamma_0 \left( \delta \right) }\left( t_{\delta }\right) - \omega^{\ast }\right\Vert _{B_{1}}=0. \end{equation*} The parameter \(\gamma\) is called the regularization parameter and the procedure of constructing the approximate solution \(\omega_{\gamma( \delta)} = R_{\gamma(\delta) }(t_{\delta })\) is called the regularization procedure, or simply regularization.

One can use different regularization procedures for the same classification problem. The regularization parameter \(\gamma\) can be even the vector of regularization parameters depending on number of iterations in the used classification method, the tolerance chosen by the user, number of classes, etc..

For two Banach spaces \(B_{1}\) and \(B_{2}\) let \(Q\) be another space, \(Q\subset B_{1}\) as a set and \(\overline{Q}=B_{1}\). In addition, we assume that \(Q\) is compactly embedded in \(B_{1}.\) Let \(G\subset B_{1}\) be the closure of an open set\(.\) Consider a continuous one-to-one function \(y:G\rightarrow B_{2}.\) Our goal is to solve

Let To find an approximate solution of equation (53), we construct the Tikhonov regularization functional \(J_{\gamma}(\omega),\) \begin{equation*} J_{\gamma}: G\rightarrow \mathbb{R}, \end{equation*} where \(\gamma = \gamma(\delta) >0\) is a small regularization parameter.

The regularization term \(\frac{\gamma}{2} \psi(\omega) \) encodes a priori available information about the unknown solution such that sparsity, smoothness, monotonicity, etc... Regularization term \( \frac{\gamma}{2} \psi(\omega) \) can be chosen in different norms, for example:

• \( \frac{\gamma}{2} \psi(\omega) = \frac{\gamma}{2} \| \omega \|^p_{L^p}, ~~ 1 \leq p \leq 2\).
• \(\frac{\gamma}{2} \psi(\omega) = \frac{\gamma}{2} \| \omega \|_{TV}\), TV means ``total variation''.
• \( \frac{\gamma}{2} \psi(\omega) = \frac{\gamma }{2} \| \omega \|_{BV}\), BV means ``bounded variation'', a real-valued function whose TV is bounded (finite).
• \( \frac{\gamma}{2} \psi(\omega) = \frac{\gamma }{2} \|\omega\|_{H^1}^2\).
• \( \frac{\gamma}{2} \psi(\omega) = \frac{\gamma }{2}( \|\omega \|_{L^1} + \| \omega \|^2_{L^2})\).

We consider the following Tikhonov functional for regularized classification problem where terms \(\varphi(\omega), \psi(\omega)\) are considered in \(L_2\) norm which is the classical Banach space. In (56) \(\omega_{0}\) is a good first approximation for the exact weight function \(\omega^{\ast }\), which is called also the first guess or the first approximation. For discussion about how the first guess in the functional (56) should be chosen we refer to [15,16,24].

In this section we will discuss following rules for choosing regularization parameter in (56):

• A-priori rule (Tikhonov's regularization)
  • - For \(\| t - t^*\| \leq \delta\) a priori rule requires (see details in [15]): \begin{equation*} \lim_{\delta \rightarrow 0} \gamma(\delta) \to 0, ~ \lim_{\delta \rightarrow 0} \frac{\delta^2}{\gamma(\delta)} \to 0. \end{equation*}
• A-posteriori rules:
  • - Morozov's discrepancy principle [17,19,25]}.
  • - Balancing principle [17].

A-priori rule and Morozov's discrepancy are most popular methods for the case when there exists estimate of the noise level \(\delta\) in data \(t\). Otherwise it is recommended to use balancing principle or other a-posteriori rules presented in [15,16,17,18,19,23].

5.1. The Tikhonov's regularization

The goal of regularization is to construct sequences \(\left\{ \gamma\left( \delta _{k}\right) \right\} ,\left\{ \omega_{\gamma \left( \delta _{k}\right) }\right\} \) in a stable way so that \begin{equation*} \lim_{k\rightarrow \infty }\left\Vert \omega_{\gamma \left( \delta _{k}\right) }- \omega^{\ast }\right\Vert _{B_{1}}=0, \end{equation*} where a sequence \(\left\{ \delta _{k}\right\} _{k=1}^{\infty }\) is such that Using (54) and (56), we obtain Let \begin{equation*} m_{\gamma \left( \delta _{k}\right) }=\inf_{G}J_{\gamma \left( \delta _{k}\right) }\left( \omega\right) . \end{equation*} By (59) \begin{equation*} m_{\gamma \left( \delta _{k}\right) } \leq \frac{\delta _{k}^{2}}{2}+\frac{% \gamma \left( \delta _{k}\right) }{2}\left\Vert \omega^{\ast } - \omega_{0}\right\Vert_{Q}^{2}. \end{equation*} Hence, there exists a point \(\omega_{\gamma \left( \delta _{k}\right) }\in G\) such that Thus, by (56) and (60) From (61) follows that Using (63) and then (60) one can obtain from what follows that Suppose that Then (65) implies that the sequence \(\left\{ \omega_{\gamma \left( \delta _{k}\right) }\right\} \subset G\subseteq Q\) is bounded in the norm of the space \(Q.\) Since \(Q\) is compactly embedded in \(B_{1},\) then there exists a sub-sequence of the sequence \(\left\{ \omega_{\gamma \left( \delta _{k}\right) }\right\} \) which converges in the norm of the space \(B_{1}.\)

To ensure (66) one can choose, for example

Other choices of \(\gamma\) which satisfy conditions (66) are also possible.

In [15,20] was proposed following iterative update of the regularization parameters \({\gamma_k}\) which satisfy conditions (66):

where \(\gamma_0\) can be computed as in (67).

5.2. Morozov's discrepancy principle

The principle determines the regularization parameter \(\gamma=\gamma(\delta)\) in (56) such that where \(c_m \geq 1\) is a constant. Relaxed version of a discrepancy principle is: for some constants \(1 \leq c_{m,1} \leq c_{m,2}\). The main feature of the principle is that the computed weight function \(\omega_{\gamma(\delta)}\) can't be more accurate than the residual \(\| y(\omega_{\gamma(\delta)}) - t \|\).

For the Tikhonov functional

the value function \(F(\gamma): \mathbb{R}^+ \to \mathbb{R}\) is defined accordingly to [23] as If there exists \(F_\gamma'(\gamma)\) at \(\gamma >0\) then from (71) and (72) follows that Since \( F_{\gamma}'(\gamma) = \psi'(\omega) = \bar{\psi}(\gamma)\) then from (73) follows The main idea of the principle is to compute discrepancy \(\bar{\varphi}(\gamma)\) using the value function \(F(\gamma)\) and then approximate \(F(\gamma)\) via model functions. If \( \bar{\psi}(\gamma) \in C(\gamma)\) then the discrepancy equation (69) can be rewritten as The goal is to solve (75) for \(\gamma\). Main methods for solution of (75) are the model function approach and a quasi-Newton method presented in details in [17].

5.3. Balancing principle

The balancing principle (or Lepskii, see [26,27]) finds \(\gamma > 0\) such that following expression is fulfilled where \(C = a_0/a_1\) is determined by the statistical a priori knowledge from shape parameters in Gamma distributions [17]. When \(\gamma=1\) the method is called zero crossing method, see details in [28]. In [17] for computing \(\gamma\) is proposed the fixed point algorithm 4. Convergence of this algorithm is also analyzed in [17].

6. Numerical results

In this section are presented several examples which show performance and effectiveness of least squares, perceptron and WINNOW algorithms for classification. We note that all classification algorithms considered here doesn't include regularization.

6.1. Test 1

In this test the goal is to compute decision boundaries for two linearly separated classes using least squares classification. Points in these classes are generated randomly by the linear function \(y = 1.2 - 0.5x\) on different input intervals for \(x\). Then the random noise \(\delta\) is added to the data \(y(x)\) as

where \(\alpha \in (1, 1)\) is randomly distributed number and \(\delta \in [0, 1]\) is the noise level. Then obtained points are classified such that the target function for classification is defined as Figures 1 present classification performed via least squares minimization \(\min_\omega \| A\omega -y \|_2^2\) for the linear model function (23) with target values \(t\) given by (78) and elements of the design matrix \(A\) given by (25). Using these figures we observe that the least squares can be used successfully for classification when it is known that classes are linearly separated.

6.2. Test 2

Here we present examples of performance of the perceptron learning algorithm for classification of two linearly separated classes. Data for analysis in these examples are generated similarly as in the Test 1. Figures 3 present classification of two classes in the perceptron algorithm with three weights. Figures 4 show classification of two classes using the second order polynomial function 46 in the perceptron algorithm with six weights. We note that the red and blue lines presented in Figures 4 are classification boundaries computed via (51). Figures 5 present comparison of linear perceptron and WINNOW algorithms for separation of two classes. Again, all these figures show that perceptron and WINNOW algorithms can be successfully used for separation of linearly separated classes.

6.3. Test 3

This test shows performance of using the second order polynomial function (46) in the perceptron algorithm for classification of the segmented solution. The classification problem in this example is formulated as follows:

• Given the computed solution of the Poisson's equation \(\triangle u = f\) with homogeneous boundary conditions \(u=0\) on the unit square, classify the discrete solution \(u_h\) into two classes such that the target function for classification is defined as (see middle figures of Figure 6:
  \begin{equation} \label{test3_1} t_i = \left \{ \begin{array}{lll} 1, & \textrm{\({u_h}_i > 4\)} & \textrm{(yellow points)}, \\ 0, & \textrm{otherwise} & \textrm{(blue points)}. \\ \end{array} \right. \end{equation}

  (79)
• Use the second order polynomial function (46) in the perceptron algorithm to compute decision boundaries.

Details about setup and numerical method to compute solution of Poisson's equation are presented in the example 8.1.3 of [20]. Figure 6 presents classification of the computed solution for Poisson's equation on different meshes using second order polynomial in classification algorithm. The computed solution \(u_h\) of the Poisson's equation on different meshes is presented on the left figures of Figure 6. The middle figures show segmentation of the obtained solution satisfying (79). The right figures of Figure 6 show results of applying the second order polynomial function (46) in the perceptron algorithm for classification of the computed solution \(u_h\) with target function (79). We observe that in this particular example computed decision lines correctly separate two classes even if these classes are not linearly separable.

6.4. Test 4

In this test we show performance of linear least squares together with linear and quadratic perceptron algorithms for classification of experimental data sets: database of grey seals [13] and Iris data set [12]. Matlab code to run these simulations is available for download from the link provided in [13].

Figure 7-a) shows classification of seal length and seal weight depending on the year. Figure 7-b) shows classification of seal length and seal thickness depending on the seal weight. We observe that classes on Figure 7-a) are not linearly separable and the best algorithm which separates both classes well, is the least squares. In this example, the linear and quadratic perceptron have not classified data correctly and actually, these algorithms have not converged and stopped when the maximal number of iterations (\(10^8\)) was reached. As soon as classes become linearly separated, all algorithms show good performance and computes almost the same separation lines, see Figure 7-b).

The same conclusion is obtained from separation of Iris data set [12]. Figures 8 show decision lines computed by least squares, linear and quadratic perceptron algorithms. Since all classes of Iris data set are linearly separable, all classification algorithms separate data correctly.

7. Conclusion

We have presented regularized and non-regularized perceptron learning and least squares algorithms for classification problems as well as discussed main a-priori and a-posteriori Tikhonov's regularization rules for choosing the regularization parameter. The Fréchet derivatives for least squares and perceptron algorithms are also rigorously derived.

The future work can be related to computation of miss-classification rates which can be done similarly with works [2,29], as well as to study of classification problem using regularized linear regression, perceptron learning and WINNOW algorithms. Other classification algorithms such that regularized SVM and kernel methods can be also analyzed. Testing of all algorithms on different benchmarks as well as extension to multiclass case should be also investigated.

acknowledgments

The research is supported by the Swedish Research Council grant VR 2018-03661.

conflictofinterests

The author declares no conflict of interest.

1. Introduction

Let \[A_n=\sum_{k=0}^nq^{k^2}\binom{n}{k},\] \[ B_n=\sum_{j\in\mathbb{Z}}(-1)^jq^{\frac{j(5j-1)}{2}}\binom{2n}{n-2j},\] \[ C_n=\sum_{k=0}^nq^{k^2+k}\binom{n}{k},\] and \[ D_n=\sum_{j\in\mathbb{Z}}(-1)^jq^{\frac{j(5j-3)}{2}}\binom{2n+1}{n+1-2j},\] where \(\binom{n}{k}\) is a \(q\)-binomial coefficients [2], defined by \begin{equation*} \binom{n}{k}:=\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}\quad\text{with}\quad (x;q)_m:=(1-x)(1-xq)\dots(1-xq^{m-1}). \end{equation*} This notation is the most common one for \(q\)-binomial coefficients, and there is no danger to mix them up with Stirling cycle numbers, as they don't appear in this paper. When the need arises to distinguish the \(q\)-parameter, the notation \(\binom{n}{k}_q\) is used. For the reader's convenience, the basic recursions will be given here: \begin{equation*} \binom{n}{k}=q^k\binom{n-1}{k}+\binom{n-1}{k-1},\quad\text{and}\quad \binom{n}{k}=\binom{n-1}{k}+q^{n-k}\binom{n-1}{k-1}. \end{equation*}

Bressoud [3] proved that \(A_n=B_n\) and \(C_n=D_n\) and that taking the limit \(n\to\infty\) leads to the celebrated Rogers-Ramanujan identities. Since it is well documented in the literature how to take this limit we will not repeat this here and concentrate on the polynomial identities. The Bressoud polynomials \(A_n\) and \(C_n\) are not the only finitizations of the celebrated Rogers-Ramanujan identities, but arguably the simplest and prettiest. More information about finite versions of Rogers-Ramanujan type identities can be found in the encyclopedic paper [1].

Chapman [4] found a simple and elementary approach, and it was used in [2] almost without change. A different simple proof was provided by Cigler [5] a few years later.
Chapman's method consists in showing that both sides of the identity satisfy the same recursion. This recursion is, however, in two variables. Also, auxiliary sequences needed to be introduced.
Here, we use a different (although related) recursion that depends only on one variable, and requires no auxiliary sequences.
It is easy to check that the first two values of the sequences also coincide, so that the sequences themselves coincide.
There are other approaches to deal with Bressoud's polynomials and extensions, like [6]. Here, we try to make everything as simple and elementary as possible.
In a final section, we apply the same machinery to the so-called Santos polynomials [7]. They belong to another pair of Rogers-Ramanujan type identities, and there is hope that even more such polynomials can be treated along the lines of this note, since Zeilberger's algorithm is helpful to establish the relevant recursions.

2. The first identity

It is a routine computation to verify that Finding this and related relations depends either on Zeilberger's algorithm or an approach of creating a sufficient amount of data and spotting patterns. This is to some extent the fruit of year-long experience with experimentation. Multiplying (1) by \(q^{k^2}\) and summing over nonnegative integers \(k\) leads to the recursion \begin{equation*} A_n-(1+q-q^n+q^{2n-1})A_{n-1}+q(1-q^{n-1})A_{n-2}=0. \end{equation*} It is more of a challenge to show the recursion which we will do now. Cigler [5] used the \(q\)-Chu-Vandermonde convolution to expand \begin{equation*} \binom{2n}{n-2j}= \sum_kq^{(k-j)(k+j)}\binom{n}{k-j}\binom{n}{k+j} \end{equation*} and consequently \begin{equation*} B_n=\sum_kq^{k^2}f(n,k) \end{equation*} with \begin{align*} f(n,k):=\sum_{j}(-1)^jq^{\frac{j(3j-1)}{2}}\binom{n}{k-j}\binom{n}{k+j}. \end{align*} This identity has (of course) a long history. This identity is at the core of the analysis of Durfee squares [8], but can also be derived from classical \(q\)-hypergeometric identities (\(q\)-Dougall, \(q\)-Dixon, \(\dots\)), as already pointed out by Cigler himself, with a reference to Warnaar [6]. It is, however, quite striking, that this polynomial identity can be used in elementary proofs related to Bressoud's polynomials. One can actually compute \(f(n,k)=\binom{n}{k}\), but this will play no role in our proof. We want to demonstrate that one only needs simple recursions for \(f(n,k)\) that appear already in [5] but are repeated here for completeness. For that, only the recursions for the \(q\)-binomial coefficients are needed. (These recursions are the two standard recursions that prove \(f(n,k)=\binom{n}{k}\) anyway.) The reason to keep this level of simplicity is that in other instances of polynomial recursions it might not be so easy to get a simple form. We start with \begin{align*} \sum_j(-1)^jq^{\frac{j(3j-1)}{2}}\binom{n-1}{k-j}\binom{n}{k+j} &=\sum_j(-1)^jq^{\frac{j(3j-1)}{2}}\binom{n-1}{k-j}\binom{n-1}{k+j}+q^{n-k}\sum_j(-1)^jq^{\frac{3j(j-1)}{2}}\binom{n-1}{k-j}\binom{n-1}{k+j-1}\\ &=f(n-1,k), \end{align*} since the last sum, on the substitution \(j\to-j+1\), turns into its own negative. Similarly, \begin{align*} \sum_j(-1)^jq^{\frac{j(3j+1)}{2}}\binom{n-1}{k-j}\binom{n}{k+j+1} &=q^{k+1}\sum_j(-1)^jq^{\frac{3j(j+1)}{2}}\binom{n-1}{k-j}\binom{n-1}{k+j+1}+\sum_j(-1)^jq^{\frac{j(3j+1)}{2}}\binom{n-1}{k-j}\binom{n-1}{k+j}\\ &=f(n-1,k), \end{align*} by the same reasoning. Therefore \begin{align*} f(n,k)&=\sum_{j}(-1)^jq^{\frac{j(3j-1)}{2}}\binom{n}{k+j}\bigg(\binom{n-1}{k-j}+q^{n-k+j}\binom{n-1}{k-j-1}\bigg)\\ &=f(n-1,k)+q^{n-k}\sum_{j}(-1)^jq^{\frac{j(3j+1)}{2}}\binom{n}{k+j}\binom{n-1}{k-j-1}\\ &=f(n-1,k)+q^{n-k}f(n-1,k-1). \end{align*} A very similar computation leads to Now we can prove that the sequence \(B_n\) satisfies the recursion (2), which means that \begin{align*} \sum_kq^{k^2}f(n,k)-(1+q-q^n)\sum_kq^{k^2}f(n-1,k)-q^{2n}\sum_kq^{k^2-2k}f(n-1,k-1)+q(1-q^{n-1})\sum_kq^{k^2}f(n-2,k)=0. \end{align*} We claim that even \begin{align*} f(n,k)-(1+q-q^n)f(n-1,k)-q^{2n-2k}f(n-1,k-1)+q(1-q^{n-1})f(n-2,k)=0. \end{align*} Using the recursion (3), this is equivalent to \begin{align*} &q^{n-k}f(n-1,k-1)-(q-q^n)f(n-1,k)-q^{2n-2k}f(n-1,k-1)+q(1-q^{n-1})f(n-2,k)\\ &=(q^{n-k}-q^{2n-2k})f(n-1,k-1)-(q-q^n)(f(n-1,k)-f(n-2,k))\\ &=(q^{n-k}-q^{2n-2k})f(n-1,k-1)-(q-q^n)q^{n-k-1}f(n-2,k-1)\\&=0. \end{align*} This may be further reduced to \begin{align*} &(1-q^{n-k})f(n-1,k-1)-(1-q^{n-1})f(n-2,k-1)\\ &=f(n-1,k-1)-f(n-2,k-1)-q^{n-k}(f(n-1,k-1)-q^{k-1}f(n-2,k-1))\\ &=q^{n-k}f(n-2,k-2)-q^{n-k}f(n-2,k-2)=0, \end{align*} which is now obvious.

Remark 1. The two coupled recursions appearing in [4] can be transformed into the recursion (2). This goes as follows, starting from \begin{align*} B_n&=B_{n-1}+q^nD_{n-1},\\ D_n-q^nB_n&=(1-q^n)D_{n-1}. \end{align*} Eliminating \(D_{n-1}\) from the second equation and replacing it in the first equation leads to \begin{align*} (1-q^n)B_n&=(1-q^n)B_{n-1}+q^nD_n-q^{2n}B_n\\ &=(1-q^n)B_{n-1}+\frac1q(B_{n+1}-B_n)-q^{2n}B_n. \end{align*} Rearranging this leads to \begin{equation*} B_{n+1}=(q-q^{n+1}+1+q^{2n+1})B_n-q(1-q^n)B_{n-1}. \end{equation*} Replacing \(n\) by \(n-1\) leads to the recursion (2).

The approach in the present paper is to find the second order recursion (in only one variable) directly, which will be used in the sequel for the second identity, as well as for the Santos-polynomials. Fortunately, the \(q\)-Zeilberger algorithm helps to find it if one does not see it otherwise. In the instance of Santos-polynomials, the recursions (4), (5) are readily found with a computer, but an elimination using auxiliary sequences would be a more elaborate process.

3. The second identity

From (1) we get \begin{equation*} \binom{n}{k}-(1+q-q^n)\binom{n-1}{k}-q^{2n-2k}\binom{n-1}{k-1}+q(1-q^{n-1})\binom{n-2}{k}=0, \end{equation*} multiplying this by \(q^{k^2+k}\) and summing over all nonnegative integers \(k\) we are led to the recursion \begin{equation*} C_n-(1+q-q^n+q^{2n})C_{n-1}+q(1-q^{n-1})C_{n-2}=0. \end{equation*} Now we will deduce the recursion \begin{equation*} D_n-(1+q-q^n+q^{2n})D_{n-1}+q(1-q^{n-1})D_{n-2}=0 \end{equation*} as well. The \(q\)-Chu-Vandermonde formula leads to \begin{equation*} \binom{2n+1}{n-2j}=\sum_k q^{k^2+k-j^2+j}\binom{n+1}{k+1-j}\binom{n}{k+j}, \end{equation*} and therefore \begin{align*} D_n&=\sum_{j}(-1)^jq^{\frac{j(5j-3)}{2}}\sum_k q^{k^2+k-j^2+j}\binom{n+1}{k+1-j}\binom{n}{k+j}\\ &=\sum_k q^{k^2+k}\sum_{j}(-1)^jq^{\frac{j(3j-1)}{2}}\binom{n+1}{k+1-j}\binom{n}{k+j}\\ &=\sum_k q^{k^2+k}f(n,k). \end{align*} From (1), viz. \begin{align*} f(n,k)-(1+q-q^n)f(n-1,k)-q^{2n-2k}f(n-1,k-1)+q(1-q^{n-1})f(n-2,k)=0 \end{align*} we get, upon multiplication with \(q^{k^2+k}\) and summing over all nonnegative integers \(k\) \begin{align*} D_n-(1+q-q^n)D_{n-1}-q^{2n}D_{n-1}+q(1-q^{n-1})D_{n-2}=0, \end{align*} as claimed.

4. Santos polynomials

The Santos polynomials are defined as \begin{equation*} S_n:=\sum_{0\le 2k\le n}q^{2k^2}\binom{n}{2k}. \end{equation*} The are used to prove identities A.39 and A.38 from the list [1]. We start from Multiplying this by \(q^{2k^2}\) and summing, we find the recursion \begin{equation*} S_{n+2}-(1+q)S_{n+1}+(q-q^{2n+2})S_n=0. \end{equation*} (Originally, it was found using Zeilberger's \(q\)-EKHAD algorithm.) The alternative form for the Santos polynomials, as to be shown, is \begin{equation*} \overline{S}_n=\sum_{j}q^{4j^2-j}\binom{n}{\lfloor\tfrac{n+1}{2} \rfloor-2j}. \end{equation*} The \(q\)-Chu-Vandermonde formula leads to \begin{equation*} \binom{n}{\lfloor\tfrac{n+1}{2} \rfloor-2j}=\sum_k \binom{\lceil\tfrac{n}{2} \rceil}{k+j}\binom{\lfloor\tfrac{n}{2} \rfloor}{k-j} q^{2k^2-2j^2}. \end{equation*} Therefore \begin{align*} \overline{S}_n&=\sum_kq^{2k^2}g(n,k) \end{align*} with \begin{equation*} g(n,k)=\sum_{j}q^{2j^2-j} \binom{\lceil\tfrac{n}{2} \rceil}{k+j}\binom{\lfloor\tfrac{n}{2} \rfloor}{k-j}. \end{equation*} Using only the recursions for the \(q\)-binomial coefficients, a tedious computation leads to \begin{equation*} g(n+2,k)-(1+q)g(n+1,k)+q\,g(n,k)-q^{2n+2-2k}g(n,k-1)=0. \end{equation*} Multiplying this by \(q^{2k^2}\) and summing over all nonnegative integers \(k\), we find the recursion \begin{equation*} \overline{S}_{n+2}-(1+q)\overline{S}_{n+1}+(q-q^{2n+2})\overline{S}_n=0. \end{equation*} Since the recursion for \(g(n,k)\) defines, together with some initial conditions, the sequence uniquely, this also shows that \(g(n,k)=\binom n{2k}\).
There is a second family of Santos polynomials, defined by \begin{equation*} T_n:=\sum_{0\le 2k+1\le n}q^{2k^2+2k}\binom{n}{2k+1}. \end{equation*} We start from Multiplying this by \(q^{2k^2+2k}\) and summing leads to \begin{equation*} T_{n+2}-(1+q)T_{n+1}+(q-q^{2n+2})T_n=0. \end{equation*} The alternative form for the second family of Santos polynomials, as to be shown, is \begin{equation*} \overline{T}_n=\sum_{j}q^{4j^2-3j}\binom{n}{\lfloor\tfrac{n+2}{2} \rfloor-2j}. \end{equation*} The \(q\)-Chu-Vandermonde formula leads to \begin{equation*} \binom{n}{\lfloor\tfrac{n+2}{2} \rfloor-2j}=\sum_k \binom{\lfloor\tfrac{n}{2} \rfloor}{k+j}\binom{\lceil\tfrac{n}{2} \rceil}{k+1-j} q^{2k^2+2k-2j^2+2j} \end{equation*} and therefore \begin{equation*} \overline{T}_n=\sum_kq^{2k^2+2k}h(n,k), \end{equation*} with \begin{equation*} h(n,k):=\sum_{j}q^{2j^2-j} \binom{\lfloor\tfrac{n}{2} \rfloor}{k+j}\binom{\lceil\tfrac{n}{2} \rceil}{k+1-j}. \end{equation*} Another elementary computation leads to \begin{equation*} h(n+2,k) -(1+q)h(n+1,k)+q\,h(n,k)-q^{2n-2k}h(n,k-1)=0. \end{equation*} Multiplying this by \(q^{2k^2+2k}\) and summing leads to \begin{equation*} \overline{T}_{n+2}-(1+q)\overline{T}_{n+1}+(q-q^{2n+2})\overline{T}_n=0, \end{equation*} as desired. Additionally, we find \(h(n,k)=\binom{n}{2k+1}\).

5. Future work

We think it would a challenge to go through Sills list [1] and make the present approach working for as many examples as possible. What we have done so far, is apart from the Bressoud polynomials, dealing with the Santos polynomials, related with A.39/A.38 from Sills (=Slater's) list. Here is another example, which seems to be interesting, related to A.79(=A.98). Consider the polynomials \begin{equation*} U_n=\sum_kq^{k^2}\binom{n+k}{2k}. \end{equation*} Zeilberger's algorithm produces \begin{equation*} \binom{n+2+k}{2k}-(1+q)\binom{n+1+k}{2k}-q^{2n+4-2k}\binom{n+k}{2k-2}+q\binom{n+k}{2k}=0, \end{equation*} respectively. The alternative form is \begin{equation*} \overline{U}_n=\sum_jq^{15j^2-j}\binom{2n}{n-5j}-\sum_jq^{15j^2-11j+2}\binom{2n}{n+2-5j}, \end{equation*} and the challenge would be to show that these polynomials also satisfy the recursion (6). One possible way to do that would be to split the sum into even indices \(2j\) and odd indices \(2j+1\). The resulting terms of the form \(\binom{2n}{n+c-10j}\) could then be split with the \(q\)-Chu-Vandermonde into two factors of the form \(\binom{n}{k+d \pm 5j}\). To work out the details might be a bit unpleasant, though.

Conflict of Interests

The author declares no conflict of interest.

1. Introduction

Let \(0< k\in\mathbb{Z}\). The middle-levels graph \(M_k\) [1] is the subgraph induced by the \(k\)-th and \((k+1)\)-th levels (formed by the \(k\)- and \((k+1)\)-subsets of \([2k+1]=\{0,\ldots,2k\}\)) in the Hasse diagram [2] of the Boolean lattice \(2^{[2k+1]}\). The dihedral group \(D_{4k+2}\) acts on \(M_k\) via translations mod \(2k+1\) (Section 4) and complemented reversals (Section 5). The sequence \(\mathcal S\) [3] {A239903} of restricted-growth strings or RGS's ([4] page 325, [5] page 224 item (u)) will be shown to unify the presentation of all \(M_k\)'s. In fact, the first \(C_k\) terms of \(\mathcal S\) will stand for the orbits of \(V(M_k)\) under natural \(D_{4k+2}\)-action, where \(C_k=\frac{(2k)!}{k!(k+1)!}\) is the \(k\)-th Catalan number [3] {A000108}. This will provide a reinterpretation (Section 10) of the Middle-Levels Theorem on the existence of Hamilton cycles in \(M_k\) [6, 7], via \(k\) -germs (Section 2) of RGS's. For the history of this theorem, [6] may be consulted, but the conjecture answered by it was learned from B. Bollob\'as on January 23, 1983, who mentioned it as a P. Erdős conjecture.

In Section 6, the cited \(D_{4k+2}\)-action on \(M_k=(V(M_k),E(M_k))\) allows to project \(M_k\) onto a quotient graph \(R_k\) whose vertices stand for the first \(C_k\) terms of \(\mathcal S\) via the lexical-matching colors [1] (or lexical colors) \(0,1,\ldots,k\) on the \(k+1\) edges incident to each vertex (Section 12). In preparation, RGS's \(\alpha\) are converted in Section 3 into \((2k+1)\)-strings \(F(\alpha)\), composed by the \(k+1\) lexical colors and \(k\) asterisks, representing all ordered (rooted) \(k\)-edge trees (Remark 1) via a ``Castling'' procedure that facilitates enumeration. These trees (encoded as \(F(\alpha)\)) are shown to represent the vertices of \(R_k\) via ``Uncastling'' procedure (Section 8).

In Section 9, the 2-factor \(W_{01}^k\) of \(R_k\) determined by the colors 0 and 1 is analyzed from the RGS-dihedral action viewpoint. From this, \(W_{01}^k\) is seen in Section 10 to morph into Hamilton cycles of \(M_k\) via symmetric differences with 6-cycles whose presentation is alternate to that of [7]. In particular, an integer sequence \({\mathcal S}_0\) is shown to exist such that, for each integer \(k>0\), the neighbors via color \(k\) of the RGS's in \(R_k\) ordered as in \(\mathcal S\) correspond to an idempotent permutation on the first \(C_k\) terms of \({\mathcal S}_0\). This and related properties hold for colors \(0,1,\ldots,k\) (Theorem 10 and Remarks 2-4) in part reflecting and extending from Observation 2 in Section 9 properties of plane trees (i.e., classes of ordered trees under root rotation). Moreover, Section 11 considers suggestive symmetry properties by reversing the designation of the roots in the ordered trees so that each lexical-color \(i\) adjacency (\(0\le i\le k\)) can be seen from the lexical-color \((k-i)\) viewpoint.

2. Restricted-Growth Strings and \(k\)-Germs

Let \(0< k\in\mathbb{Z}\). The sequence of (pairwise different) RGS's \({\mathcal S}=(\beta(0),\ldots,\beta(17),\ldots)=(0,1,10,11,12,100,101,110,111,112,120,121,122,123,1000,1001,1010,1011,\ldots)\) has the lengths of its contiguous pairs \((\beta(i-1),\beta(i))\) constant unless \(i=C_k\) for \(0< k\in\mathbb{Z}\), in which case \(\beta(i-1)=\beta(C_k-1)=12\cdots k\) and \(\beta(i)=\beta(C_k)=10^k=10\cdots 0\).

To view the continuation of the sequence \(\mathcal S\), each RGS \(\beta=\beta(m)\) will be transformed, for every \(k\in\mathbb{Z}\) such that \(k\ge\) length\((\beta)\), into a \((k-1)\)-string \(\alpha=a_{k-1}a_{k-2}\cdots a_2a_1\) by prefixing \(k-\) length\((\beta)\) zeros to \(\beta\). We say such \(\alpha\) is a \(k\)- germ. More generally, a \(k\)- germ \(\alpha\) (\(1< k\in\mathbb{Z}\)) is formally defined to be a \((k-1)\)-string \(\alpha=a_{k-1}a_{k-2}\cdots a_2a_1\) such that:

• (1) the leftmost position (called position \(k-1\)) of \(\alpha\) contains the entry \(a_{k-1}\in\{0,1\}\);
• (2) given \(1 < i< k \), the entry \(a_{i-1}\) (at position \(i-1\)) satisfies \(0\le a_{i-1}\le a_i+1\).

Every \(k\)-germ \(a_{k-1}a_{k-2}\cdots a_2a_1\) yields the \((k+1)\)-germ \(0a_{k-1}a_{k-2}\cdots\) \(a_2a_1\). A non-null RGS is obtained by stripping a \(k\)-germ \(\alpha=a_{k-1}a_{k-2}\cdots a_1\) \(\ne 00\cdots 0\) off all the null entries to the left of its leftmost position containing a 1. Such a non-null RGS is again denoted \(\alpha\). We also say that the null RGS \(\alpha=0\) corresponds to every null \(k\)-germ \(\alpha\), for \(0< k\in\mathbb{Z}\). (We use the same notations \(\alpha=\alpha(m)\) and \(\beta=\beta(m)\) to denote both a \(k\)-germ and its associated RGS). The \(k\)-germs are ordered as follows. Given 2 \(k\)-germs, say \(\alpha=a_{k-1}\cdots a_2a_1\) and \(\beta=b_{k-1}\cdots b_2b_1,\) where \(\alpha\ne \beta\), we say that \(\alpha\) precedes \(\beta\), written \(\alpha< \beta\), whenever either:

• (i) \(a_{k-1} < b_{k-1}\) or
• (ii) \(a_j=b_j\), for \(k-1\le j\le i+1\), and \(a_i < b_i\), for some \(k-1>i\ge 1\).

The resulting order on \(k\)-germs \(\alpha(m)\), (\(m\le C_k\)), corresponding biunivocally (via the assignment \(m\rightarrow\alpha(m)\)) with the natural order on \(m\), yields a listing that we call the natural (\(k\)-germ) enumeration. Note that there are exactly \(C_k\) \(k\)-germs \(\alpha=\alpha(m)< 10^k\), \(\forall k>0\).

3. Castling of ordered trees

Table 1. The Castling Procedure for \(k=2,3,4\).

\(m\)	\(\alpha\)	\(\beta\)	\(F(\beta)\)	\(i\)	\(W^i\,|\,X\,|\,Y\,|\,Z^i\)	\(W^i\,|\,Y\,|\,X\,|\,Z^i\)	\(F(\alpha)\)	\(\alpha\)
0	0	-	-	-	-	-	012**	0
1	1	0	012**	1	0\,|\,1\,|\,2*|*	0|2*|1|*	02*1*	1
0	00	-	-	-	-	-	012\,3**\,*	00
1	01	00	0123***	1	0|1|23**|*	0|23**|1|*	023**1*	01
2	10	00	0123***	2	01|2|3*|**	01|3*|2|**	013*2**	10
3	11	10	013*2**	1	0|13*|2*|*	0|2*|13*|*	02*13**	11
4	12	11	02*13**	1	0|2*1|3*|*	0|3*|2*3|*	03*2*1*	12
0	000	-	-	-	-	-	01234****	000
1	001	000	01234****	1	0|1|234***|*	0|234***|1|*	0234***1*	001
2	010	000	01234****	2	01|2|34**|**	01|34**|2|**	0134**2**	010
3	011	010	0134**2**	1	0|134**|2*|*	0|2*|134**|*	02*134***	011
4	012	011	02*134***	1	0|2*1|34**|*	0|34**|2*1|*	034**2*1*	012
5	100	000	01234****	3	012|3|4*|***	012|4*|3|***	0124*3***	100
6	101	100	0124*3***	1	0|1|24*3**|*	0|24*3**|1|*	024*3**1*	101
7	110	100	0124*3***	2	01|24*|3*|**	01|3*|24*|**	013*24***	110
8	111	110	013*24***	1	0|13*|24**|*	0|24**|13*|*	024**13**	111
9	112	111	024**13**	1	0|24**1|3*|*	0|3*|24**1|*	03*24**1*	112
10	120	110	013*24***	2	01|3*2|4*|**	01|4*|3*2|**	014*3*2**	120
11	121	120	014*3*2**	1	0|14*3*|2*|*	0|2*|14*3*|*	02*14*3**	121
12	122	121	02*14*3**	1	0|2*34*|3*|*	0|3*|2*14*|*	03*2*14**	122
13	123	122	03*2*14**	1	0|3*2*1|4*|*	0|4*|3*2*1|*	04*3*2*1*	123

Theorem 1. To each \(k\)-germ \(\alpha=a_{k-1}\cdots a_1\ne 0^{k-1}\) with rightmost entry \(a_i\ne 0\) (\(k>i\ge 1\)) corresponds a \(k\)-germ \(\beta(\alpha)=b_{k-1}\cdots b_1\!< \alpha\) with \(b_i= a_i-1\) and \(a_j=b_j\), (\(j\ne i\)). All \(k\)-germs form an ordered tree \({\mathcal T}_k\) rooted at \(0^{k-1}\), each \(k\)-germ \(\alpha\ne0^{k-1}\) with \(\beta(\alpha)\) as parent.

Proof. The statement, illustrated for \(k=2,3,4\) by means of the first 3 columns of Table 1, is straightforward. Table 1 also serves as illustration to the proof of Theorem 2, below.

By representing \({\mathcal T}_k\) with each node \(\beta\) having its children \(\alpha\) enclosed between parentheses following \(\beta\) and separating siblings with commas, we can write: $${\mathcal T}_4=000(001,010(011(012)),100(101,110(111(121)),120(121(122(123))))).$$

Theorem 2. To each \(k\)-germ \(\alpha=a_{k-1}\cdots a_1\) corresponds a \((2k+1)\)-string \(F(\alpha)=f_0f_1\cdots f_{2k}\) whose entries are both the numbers \(0,1,\ldots,k\) (once each) and \(k\) asterisks (\(*\)) and such that:

• (A) \(F(0^{k-1})=012\cdots(k-2)(k-1)k*\cdots *\);
• (B) if \(\alpha\ne 0^{k-1}\), then \(F(\alpha)\) is obtained from \(F(\beta)=F(\beta(\alpha))=h_0h_1\cdots h_{2k}\) by means of the following ``Castling Procedure'' steps:

• let \(W^i=h_0h_1\cdots h_{i-1}=f_0f_1\cdots f_{i-1}\) and \(Z^i=h_{2k-i+1}\cdots h_{2k-1}h_{2k}=f_{2k-i+1}\cdots f_{2k-1}f_{2k}\) be respectively the initial and terminal substrings of length \(i\) in \(F(\beta)\);
• let \(\Omega>0\) be the leftmost entry of the substring \(U=F(\beta)\setminus(W^i\cup Z^i)\) and consider the concatenation \(U=X|Y\), with \(Y\) starting at entry \(\Omega+1\); then, \(F(\beta)=W^i|X|Y|Z^i\);
• set \(F(\alpha)=W^i|Y|X|Z^i\).

In particular:

• (a) the leftmost entry of each \(F(\alpha)\) is \(0\); \(k*\) is a substring of \(F(\alpha)\), but \(*k\) is not;
• (b) a number to the immediate right of any \(b\in[0,k)\) in \(F(\alpha)\) is larger than \(b\);
• (c) \(W^i\) is an (ascending) number \(i\)-substring and \(Z^i\) is formed by \(i\) of the \(k\) asterisks.

Proof.

Let \(\alpha=a_{k-1}\cdots a_1\ne 0^{k-1}\) be a \(k\)-germ. In the sequence of applications of 1-3 along the path from root \(0^{k-1}\) to \(\alpha\) in \({\mathcal T}_k\), unit augmentation of \(a_i\) for larger values of \(i\), (\(0< i< k\)), must occur earlier, and then in strictly descending order of the entries \(i\) of the intermediate \(k\)-germs. As a result, the length of the inner substring \(X|Y\) is kept non-decreasing after each application. This is illustrated in Table 1, where the order of presentation of \(X\) and \(Y\) is reversed in successively decreasing steps. In the process, (a)-(c) are seen to be fulfilled.

The 3 successive subtables in Table 1 have \(C_k\) rows each, where \(C_2=2\), \(C_3=5\) and \(C_4=14\); in the subtables, the \(k\)-germs \(\alpha\) are shown both on the second and last columns via natural enumeration in the first column; the images \(F(\alpha)\) of those \(\alpha\) are on the penultimate column; the remaining columns in the table are filled, from the second row on, as follows:

• (i) \(\beta=\beta(\alpha)\), arising in Theorem 1
• (ii) \(F(\beta)\), taken from the penultimate column in the previous row;
• (iii) the length \(i\) of \(W^i\) and \(Z^i\) (\(k-1\ge i\ge 1\));
• (iv) the decomposition \(W^i|Y|X|Z^i\) of \(F(\beta)\);
• (v) the decomposition \(W^i|X|Y|Z^i\) of \(F(\alpha)\), re-concatenated in the following, penultimate, column as \(F(\alpha)\), with \(\alpha=F^{-1}(F(\alpha))\) in the last column.

Remark 1. As in the case of \({\mathcal T}_k\) in Theorem 1, an ordered (rooted) tree [7] is a tree \(T\) with:

• (a) a specified node \(v_0\) as the root of \(T\);
• (b) an embedding of \(T\) into the plane with \(v_0\) on top;
• (c) the edges between the \(j\)- and \((j+1)\)-levels of \(T\) (formed by the nodes at distance \(j\) and \((j+1)\) from \(v_0\), where \(0\le j< \) height\((T)\)) having (parent) nodes at the \(j\)-level above their children at the \((j+1)\)-level;
• (d) the children in (c) ordered in a left-to-right fashion.

Each \(k\)-edge ordered tree \(T\) is both represented by a \(k\)-germ \(\alpha\) and by its associated \((2k+1)\)-string \(F(\alpha)\), so we write \(T=T_\alpha\). In fact, we perform a depth first search (DFS, or \(\rightarrow\)DFS) on \(T\) with its vertices from \(v_0\) downward denoted as \(v_i\) (\(i=0,1,\ldots,k\)) in a right-to-left breadth-first search (\(\leftarrow\)BFS) way. Such DFS yields the claimed \(F(\alpha)\) by writing successively:

• (i) the subindex \(i\) of each \(v_i\) in the \(\rightarrow\)DFS downward appearance and
• (ii) an asterisk for the edge \(e_i\) ending at each child \(v_i\) in the \(\rightarrow\)DFS upward appearance.

Then, we write: \(F(T_\alpha)=F(\alpha).\) Now, Theorem 1 can be taken as a tree-surgery transformation from \(T_\beta\) onto \(T_\alpha\) for each \(k\)-germ \(\alpha\ne 0^{k-1}\) via the vertices \(v_i\) and edges \(e_i\) (whose parent vertices are generally reattached in different ways). This remark is used in Sections 9-11 in reinterpreting the Middle-Levels Theorem. (In Section 11, an alternate viewpoint on ordered trees taking \(a_k\) as the root instead of \(a_0\) is considered).

To each \(F(\alpha)\) corresponds a binary \(n\)-string \(\theta(\alpha)\) of weight \(k\) obtained by replacing each number in \([k]=\{0,1,\ldots,k-1\}\) by 0 and each asterisk \(*\) by 1. By attaching the entries of \(F(\alpha)\) as subscripts to the corresponding entries of \(\theta(\alpha)\), a subscripted binary \(n\)-string \(\hat{\theta}(\alpha)\) is obtained, as shown for \(k=2,3\) in the 4th column of Table 2. Let \(\aleph(\theta(\alpha))\) be given by the complemented reversal of \(\theta(\alpha)\), that is: \begin{eqnarray}\label{d2}\mbox{if }\theta(\alpha)=a_0a_1\cdots a_{2k}\mbox{, then }\aleph(\theta(\alpha ))=\bar{a}_{2k}\cdots\bar{a}_1\bar{a}_0,\end{eqnarray} where \(\bar{0}=1\) and \(\bar{1}=0\). A subscripted version \(\hat{\aleph}\) of \(\aleph\) is obtained for \(\hat{\theta}(\alpha)\), as shown in the fifth column of Table 2, with the subscripts of \(\hat{\aleph}\) reversed with respect to those of \(\aleph\). Each image of a \(k\)-germ \(\alpha\) under \(\aleph\) is an \(n\)-string of weight \(k+1\) and has the 1's indexed with numeric subscripts and the 0's indexed with the asterisk subscript. The number subscripts reappear from Section 7 on as lexical colors [1] for the graphs \(M_k\).

Table 2. Subscripted binary \(n\)-strings \(\hat{\theta}(\alpha)\) and complemented reversals via \(\aleph\) for \(k=2,3\).

\(m\)	\(\alpha\)	\(\theta(\alpha)\)	\(\hat{\theta}(\alpha)\)	\(\hat{\aleph}(\theta(\alpha))=\aleph(\hat{\theta}(\alpha))\)	\(\aleph(\theta(\alpha))\)
0	0	00011	\(0_00_10_21_*1_*\)	\(0_*0_*1_21_11_0\)	00111
1	1	00101	\(0_00_21_*0_11_*\)	\(0_*1_10_*1_21_0\)	01011
0	00	0000111	\(0_00_10_20_31_*1_*1_*\)	\(0_*0_*0_*1_31_21_11_0\)	0001111
1	01	0001101	\(0_00_20_31_*1_*0_11_*\)	\(0_*1_10_*0_*1_31_21_0\)	0100111
2	10	0001011	\(0_00_10_32_*0_11_*1_*\)	\(0_*0_*1_20_*1_31_11_0\)	0010111
3	11	0010011	\(0_00_21_*0_10_31_*1_*\)	\(0_*0_*1_31_10_*1_21_0\)	0011011
4	12	0010101	\(0_00_31_*0_21_*0_11_*\)	\(0_*1_10_*1_20_*1_31_0\)	0101011

4. Translations \(\mod n=2k+1\)

Let \(n=2k+1\). The \(n\)- cube graph \(H_n\) is the Hasse diagram of the Boolean lattice \(2^{[n ]}\) on the set \([n]=\{0,\ldots,n-1\}\). It is convenient to express each vertex \(v\) of \(H_n\) in 3 different equivalent ways:

• (a) ordered set \(A=\{a_0,a_1,\ldots,a_{j-1}\}=a_0a_1\cdots a_{j-1}\subseteq [n]\) that \(v\) represents, (\(0< j\le n\));
• (b) characteristic binary \(n\)-vector \(B_A=(b_0,b_1,\ldots,b_{n-1})\) of ordered set \(A\) in (a) above, where \(b_i=1\) if and only if \(i\in A\), (\(i\in[n]\));
• (c) polynomial \(\epsilon_A(x)=b_0+b_1x+\cdots +b_{n-1}x^{n-1}\) associated to \(B_A\) in (b) above.

Ordered set \(A\) and vector \(B_A\) in (a) and (b) respectively are written for short as \(a_0a_1\cdots a_{j-1}\) and \(b_0b_1\cdots b_{n-1}\). \(A\) is said to be the support of \(B_A\).
For each \(j\in[n]\), let \(L_j=\{A\subseteq[n]\mbox{ with }|A|=j\}\) be the \(j\)- level of \(H_n\). Then, \(M_k\) is the subgraph of \(H_n\) induced by \(L_k\cup L_{k+1}\), for \(1\le k\in\mathbb{Z}\). By viewing the elements of \(V(M_k)=L_k\cup L_{k+1}\) as polynomials, as in (c) above, a regular (i.e., free and transitive) translation mod \(n\) action \(\Upsilon'\) of \(\mathbb{Z}_n\) on \(V(M_k)\) is seen to exist, given by: \begin{eqnarray}\label{d3}\Upsilon':\mathbb{Z}_n\times V(M_k)\rightarrow V(M_k)\mbox{, with }\Upsilon'(i,v)=v(x)x^i\mbox{ (mod }1+x^n),\end{eqnarray} where \(v\in V(M_k)\) and \(i\in\mathbb{Z}_n\). Now, \(\Upsilon'\) yields a quotient graph \(M_k/\pi\) of \(M_k\), where \(\pi\) stands for the equivalence relation on \(V(M_k)\) given by: $$\epsilon_A(x)\pi\epsilon_{A'}(x)\Longleftrightarrow\exists\,i\in\mathbb{Z}\mbox{ with }\epsilon_{A'}(x)\equiv x^i\epsilon_A(x)\mbox{ (mod }1+x^n),$$ with \(A,A'\in V(M_k)\). This is to be used in the proofs of Theorems 4 and 8. Clearly, \(M_k/\pi\) is the graph whose vertices are the equivalence classes of \(V(M_k)\) under \(\pi\). Also, \(\pi\) induces a partition of \(E(M_k)\) into equivalence classes, to be taken as the edges of \(M_k/\pi\).

5. Complemented reversals

Let \((b_0b_1\cdots b_{n-1})\) denote the class of \(b_0b_1\cdots b_{n-1}\in L_i\) in \(L_i/\pi\). Let \(\rho_i:L_i\rightarrow L_i/\pi\) be the canonical projection given by assigning \(b_0b_1\cdots b_{n-1}\) to \((b_0b_1\cdots b_{n-1})\), for \(i\in\{k,k+1\}\). The definition of \(\aleph\) in display~(\ref{d2}) is easily extended to a bijection, again denoted \(\aleph\), from \(L_k\) onto \(L_{k+1}\). Let \(\aleph_\pi:L_k/\pi\rightarrow L_{k+1}/\pi\) be given by \(\aleph_\pi((b_0b_1\cdots b_{n-1}))=(\bar{b}_{n-1}\cdots\bar{b}_1\bar{b_0})\). Observe \(\aleph_\pi\) is a bijection. Notice the commutative identities \(\rho_{k+1}\aleph=\aleph_\pi\rho_k\) and \(\rho_k\aleph^{-1}=\aleph_\pi^{-1}\rho_{k+1}\).
The following geometric representations will be handy. List vertically the vertex parts \(L_k\) and \(L_{k+1}\) of \(M_k\) (resp. \(L_k/\pi\) and \(L_{k+1}/\pi\) of \(M_k/\pi\)) so as to display a splitting of \(V(M_k)=L_k\cup L_{k+1}\) (resp. \(V(M_k)/\pi=L_k/\pi\cup L_{k+1}/\pi\)) into pairs, each pair contained in a horizontal line, the 2 composing vertices of such pair equidistant from a vertical line \(\phi\) (resp. \(\phi/\pi\), depicted through \(M_2/\pi\) on the left of Figure 1, Section 6 below). In addition, we impose that each resulting horizontal vertex pair in \(M_k\) (resp. \(M_k/\pi\)) be of the form \((B_A,\aleph(B_A))\) (resp. \(((B_A),(\aleph(B_A))=\aleph_\pi((B_A)))\)), disposed from left to right at both sides of \(\phi\). A non-horizontal edge of \(M_k/\pi\) will be said to be a skew edge.

Theorem 3. To each skew edge \(e=(B_{A})(B_{A'})\) of \(M_k/\pi\) corresponds another skew edge \(\aleph_\pi((B_{A}))\aleph^{-1}_\pi((B_{A'}))\) obtained from \(e\) by reflection on the line \(\phi/\pi\). Moreover:

• (i) the skew edges of \(M_k/\pi\) appear in pairs, with the endpoints of the edges in each pair forming 2 horizontal pairs of vertices equidistant from \(\phi/\pi\);
• (ii) each horizontal edge of \(M_k/\pi\) has multiplicity equal either to 1 or to 2.

Proof. The skew edges \(B_{A}B_{A'}\) and \(\aleph^{-1}(B_{A'})\aleph(B_{A})\) of \(M_k\) are reflection of each other about \(\phi\). Their endopoints form 2 horizontal pairs \((B_{A},\aleph(B_{A'}))\) and \((\aleph^{-1}(B_{A}),B_{A'})\) of vertices. Now, \(\rho_k\) and \(\rho_{k+1}\) extend together to a covering graph map \(\rho:M_k\rightarrow M_k/\pi\), since the edges accompany the projections correspondingly, exemplified for \(k=2\) as follows: \begin{align*}\aleph((B_A))&=\aleph((00011))\\ &=\aleph(\{00011,10001,11000,01100,00110\})\\ &=\{00111,01110,11100,11001,10011\}\\ &=(00111),\\ \aleph^{-1}((B_A'))&=\aleph^{-1}((01011))\\ &=\aleph^{-1}(\{01011,10110,10110,11010,10101\})\\ &=\{00101,10010,01001,10100,01010\}\\ &=(00101).\end{align*} Here, the order of the elements in the image of class \((00011)\) (resp. \((01011)\)) mod \(\pi\) under \(\aleph\) (resp. \(\aleph^{-1}\)) are shown reversed, from right to left (cyclically between braces, continuing on the right once one reaches the leftmost brace). Such reversal holds for every \(k>2\): \begin{align*}\aleph((B_A))&=\aleph((b_0\cdots b_{2k}))\\ &=\aleph(\{b_0\cdots b_{2k},\;b_{2k}\ldots b_{2k-1},\;\ldots,\;b_1\cdots b_0\})\\ &= \{\bar{b}_{2k}\cdots\bar{b}_0,\;\bar{b}_{2k-1}\cdots\bar{b}_{2k},\;\ldots,\;\bar{b}_1\cdots\bar{b}_0\}=\\ &(\bar{b}_{2k}\cdots\bar{b}_0),\\ \aleph^{-1}((B_A'))&=\aleph^{-1}((\bar{b}'_{2k}\cdots\bar{b}'_0))\\ &=\aleph^{-1}(\{\bar{b}'_{2k}\cdots\bar{b}'_0,\;\bar{b}'_{2k-1}\cdots\bar{b}'_{2k},\;\ldots,\;\bar{b}'_1\cdots\bar{b}'_0\})\\ &= \{b'_0\cdots b'_{2k},\;b'_{2k}\cdots b'_{2k-1},\;\ldots,\;b'_1\cdots b'_0\}\\ &=(b'_0\cdots b'_{2k}),\end{align*} where \((b_0\cdots b_{2k})\in L_k/\pi\) and \((b'_0\cdots b'_{2k})\in L_{k+1}/\pi\). This establishes (i).
Every horizontal edge \(v\aleph_\pi(v)\) of \(M_k/\pi\) has \(v\in L_k/\pi\) represented by \(\bar{b}_k\cdots \bar{b}_10b_1\cdots b_k\) in \(L_k\), (so \(v=(\bar{b}_k\cdots \bar{b}_10b_1\cdots b_k)\)). There are \(2^k\) such vertices in \(L_k\) and at most \(2^k\) corresponding vertices in \(L_k/\pi\). For example, \((0^{k+1}1^k)\) and \((0(01)^k)\) are endpoints in \(L_k/\pi\) of 2 horizontal edges of \(M_k/\pi\), each. To prove that this implies (ii), we have to see that there cannot be more than 2 representatives \(\bar{b}_k\cdots \bar{b}_1b_0b_1\cdots b_k\) and \(\bar{c}_k\cdots \bar{c}_1c_0c_1\cdots c_k\) of a vertex \(v\in L_k/\pi\), with \(b_0=0=c_0\). Such a \(v\) is expressible as \(v=(d_0 \cdots b_0d_{i+1}\cdots d_{j-1}c_0\cdots d_{2k})\), with \(b_0=d_i\), \(c_0=d_j\) and \(0< j-i\le k\). Let the substring \(\sigma=d_{i+1}\cdots d_{j-1}\) be said \((j-i)\)- feasible. Let us see that every \((j-i)\)-feasible substring \(\sigma\) forces in \(L_k/\pi\) only vertices \(\omega\) leading to 2 different (parallel) horizontal edges in \(M_k/\pi\) incident to \(v\). In fact, periodic continuation mod \(n\) of \(d_0\cdots d_{2k}\) both to the right of \(d_j=c_0\) with minimal cyclic substring \(\bar{d}_{j-1}\cdots\bar{d}_{i+1}1d_{i+1}\cdots d_{j-1}0=P_r\) and to the left of \(d_i=b_0\) with minimal cyclic substring \(0d_{i+1}\cdots d_{j-1}1\bar{d}_{j-1}\cdots\bar{d}_{i+1}=P_\phi\) yields a 2-way infinite string that winds up onto a class \((d_0\cdots d_{2k})\) containing such an \(\omega\). For example, some pairs of feasible substrings \(\sigma\) and resulting vertices \(\omega\) are: \begin{align*}(\sigma,\omega)=&(\emptyset,({ o}{ o}1)),\;\;(0,({ o}0{ o}11)),\;\;(1,({ o}1{ o})),\;\;(0^2,({ o}00{ o}111)),\;\;(01,({ o}01{ o}011)),\;\;(1^2,\,{ o}11{ o}0)),\\&(0^3,{ o}000{ o}1111)),\;\; (010,({ o}010{ o}101101)),\;\;(01^2,({ o}011{ o})),\;\;(101,({ o}101{ o})),\;\; (1^3,({ o}111{ o}00)),\end{align*} with `o' replacing \(b_0=0\) and \(c_0=0\), and where \(k=\lfloor\frac{n}{2}\rfloor\) has successive values \(k=1,2,1,3,3,2,4,5,2,2,3\). If \(\sigma\) is a feasible substring and \(\bar{\sigma}\) is its complemented reversal via \(\aleph\), then the possible symmetric substrings \(P_\phi\sigma P_r\) about \({o}\sigma{o}=0\sigma 0\) in a vertex \(v\) of \(L_k/\pi\) are in order of ascending length: $$\begin{array}{c} 0\sigma 0,\\{\bar{\sigma}0\sigma 0\bar{\sigma},}\\ 1\bar{\sigma}0\sigma 0\bar{\sigma}1,\\ {\sigma 1\bar{\sigma}0\sigma 0\bar{\sigma}1\sigma,}\\ {0\sigma 1\bar{\sigma}0\sigma 0\bar{\sigma}1\sigma 0,}\\ {\bar{\sigma}0\sigma 1\bar{\sigma}0\sigma 0\bar{\sigma}1\sigma 0\bar{\sigma},}\\ {1\bar{\sigma}0\sigma 1\bar{\sigma}0\sigma 0\bar{\sigma}1\sigma 0\bar{\sigma}1,}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots,} \end{array}$$ where we use again `0' instead of `o' for the entries immediately preceding and following the shown central copy of \(\sigma\). The lateral periods of \(P_r\) and \(P_\phi\) determine each one horizontal edge at \(v\) in \(M_k/\pi\) up to returning to \(b_0\) or \(c_0\), so no entry \(e_0=0\) of \((d_0\cdots d_{2k})\) other than \(b_0\) or \(c_0\) happens such that \((d_0\cdots d_{2k})\) has a third representative \(\bar{e}_k\cdots \bar{e}_10e_1\cdots e_k\) (besides \(\bar{b}_k\cdots \bar{b}_10b_1\cdots b_k\) and \(\bar{c}_k\cdots \bar{c}_10c_1\cdots c_k\)). Thus, those 2 horizontal edges are produced solely from the feasible substrings \(d_{i+1}\cdots d_{j-1}\) characterized above.

To illustrate Theorem 3, let \(1< h< n\) in \(\mathbb{Z}\) be such that \(\gcd(h,n)=1\) and let \(\lambda_h:L_k/\pi\rightarrow L_k/\pi\) be given by \(\lambda_h((a_0a_1\cdots a_n))\rightarrow(a_0a_ha_{2h}\cdots a_{n-2h}a_{n-h}).\) For each such \(h\le k\), there is at least one \(h\)-feasible substring \(\sigma\) and a resulting associated vertex \(v\in L_k/\pi\) as in the proof of Theorem 3. For example, starting at \(v=(0^{k+1}1^k)\in L_k/\pi\) and applying \(\lambda_h\) repeatedly produces a number of such vertices \(v\in L_k/\pi\). If we assume \(h=2h'\) with \(h'\in\mathbb{Z}\), then an \(h\)-feasible substring \(\sigma\) has the form \(\sigma=\bar{a}_1\cdots\bar{a}_{h'}a_{h'}\cdots a_1\), so there are at least \(2^{h'}=2^{\frac{h}{2}}\) such \(h\)-feasible substrings.

6. Dihedral quotients

An involution of a graph \(G\) is a graph map \(\aleph:G\rightarrow G\) such that \(\aleph^2\) is the identity. If \(G\) has an involution, an \(\aleph\)-folding of \(G\) is a graph \(H\), possibly with loops, whose vertices \(v'\) and edges or loops \(e'\) are respectively of the form \(v'=\{v,\aleph(v)\}\) and \(e'=\{e,\aleph(e)\}\), where \(v\in V(G)\) and \(e\in E(G)\); \(e\) has endvertices \(v\) and \(\aleph(v)\) if and only if \(\{e,\aleph(e)\}\) is a loop of \(G\).
Note that both maps \(\aleph:M_k\rightarrow M_k\) and \(\aleph_\pi:M_k/\pi\rightarrow M_k/\pi\) in Section 5 are involutions. Let \(\langle B_A\rangle\) denote each horizontal pair \(\{(B_A),\aleph_\pi((B_A))\}\) (as in Theorem 3) of \(M_k/\pi\), where \(|A|=k\). An \(\aleph\)-folding \(R_k\) of \(M_k/\pi\) is obtained whose vertices are the pairs \(\langle B_A\rangle\) and having:

• (1) an edge \(\langle B_A\rangle\langle B_{A'}\rangle\) per skew-edge pair \(\{(B_A)\aleph_\pi((B_{A'})),(B_{A'})\aleph_\pi((B_A))\};\)
• (2) a loop at \(\langle B_A\rangle\) per horizontal edge \((B_A)\aleph_\pi((B_A))\); because of Theorem 3, there may be up to 2 loops at each vertex of \(R_k\).

Theorem 4. \(R_k\) is a quotient graph of \(M_k\) under an action \(\Upsilon:D_{2n}\times M_k\rightarrow M_k\).

Proof. \(D_{2n}\) is the semidirect product \(\mathbb{Z}_n\rtimes_\varrho\mathbb{Z}_2\) via the group homomorphism \(\varrho:\mathbb{Z}_2\rightarrow\mbox{Aut}(\mathbb{Z}_n)\), where \(\varrho(0)\) is the identity and \(\varrho(1)\) is the automorphism \(i\rightarrow(n-i)\), \(\forall i\in\mathbb{Z}_n\). If \(*:D_{2n}\times D_{2n}\rightarrow D_{2n}\) indicates group multiplication and \(i_1,i_2\in\mathbb{Z}_n\), then \((i_1,0)*(i_2,j)=(i_1+i_2,j)\) and \((i_1,1)*(i_2,j)=(i_1-i_2,\bar{j})\), for \(j\in\mathbb{Z}_2\). Set \(\Upsilon((i,j),v)=\Upsilon'(i,\aleph^j(v))\), \(\forall i\in\mathbb{Z}_n, \forall j\in\mathbb{Z}_2\), where \(\Upsilon'\) is as in display (2). Then, \(\Upsilon\) is a well-defined \(D_{2n}\)-action on \(M_k\). By writing \((i,j)\cdot v=\Upsilon((i,j),v)\) and \(v=a_0\cdots a_{2k}\), we have \((i,0)\cdot v=a_{n-i+1}\cdots a_{2k}a_0\cdots a_{n-i}=v'\) and \((0,1)\cdot v'=\bar{a}_{i-1}\cdots \bar{a}_0\bar{a}_{2k}\cdots\bar{a}_i=(n-i,1)\cdot v=((0,1)*(i,0))\cdot v\), leading to the compatibility condition \(((i,j)*(i',j'))\cdot v=(i,j)\cdot ((i',j')\cdot v)\).

Theorem 4 yields a graph projection \(\gamma_k:M_k/\pi\rightarrow R_k\) for the action \(\Upsilon\), given for \(k=2\) in Figure 1. In fact, \(\gamma_2\) is associated with reflection of \(M_2/\pi\) about the dashed vertical symmetry axis \(\phi/\pi\) so that \(R_2\) (containing 2 vertices and one edge between them, with each vertex incident to 2 loops) is given as its image. Both the representations of \(M_2/\pi\) and \(R_2\) in the figure have their edges indicated with colors 0,1,2, as arising inSection 7.

7. Lexical procedure

Let \(P_{k+1}\) be the subgraph of the unit-distance graph of \(\mathbb{R}\) (the real line) induced by the set \([k+1]=\{0,\ldots,k\}\). We draw the grid \(\Gamma=P_{k+1}\square P_{k+1}\) in the plane \(\mathbb{R}^2\) with a diagonal \(\partial\) traced from the lower-left vertex \((0,0)\) to the upper-right vertex \((k,k)\). For each \(v\in L_k/\pi\), there are \(k+1\) \(n\)-tuples of the form \(b_0b_1\cdots b_{n-1}=0b_1\cdots b_{n-1}\) that represent \(v\) with \(b_0=0\). For each such \(n\)-tuple, we construct a \(2k\)-path \(D\) in \(\Gamma\) from \((0,0)\) to \((k,k)\) in \(2k\) steps indexed from \(i=0\) to \(i=2k-1\). This leads to a lexical edge-coloring implicit in [1]; see the following statement and Figure 2 (Section 8), containing examples of such a \(2k\)-path \(D\) in thick trace.

Theorem 5. [1] Each \(v\in L_k/\pi\) has its \(k+1\) incident edges assigned colors \(0,1,\ldots,k\) by means of the following ``Lexical Procedure'', where \(0\le i\in\mathbb{Z}\), \(w\in V(\Gamma)\) and \(D\) is a path in \(\Gamma\). Initially, let \(i=0\), \(w=(0,0)\) and \(D\) contain solely the vertex \(w\). Repeat \(2k\) times the following sequence of steps (1)-(3), and then perform once the final steps (4)-(5):

• (1) If \(b_i=0\), then set \(w':=w+(1,0)\); otherwise, set \(w':=w+(0,1)\).
• (2) Reset \(V(D):=v(D)\cup\{w'\}\), \(E(D):=E(D)\cup\{ww'\}\), \(i:=i+1\) and \(w:=w'\).
• (3) If \(w\ne(k,k)\), or equivalently, if \(i< 2k\), then go back to step (1).
• (4) Set \(\check{v}\in L_{k+1}/\pi\) to be the vertex of \(M_k/\pi\) adjacent to \(v\) and obtained from its representative \(n\)-tuple \(b_0b_1\cdots\) \(b_{n-1}=0b_1\cdots b_{n-1}\) by replacing the entry \(b_0\) by \(\bar{b}_0=1\) in \(\check{v}\), keeping the entries \(b_i\) of \(v\) unchanged in \(\check{v}\) for \(i>0\).
• (5) Set the color of the edge \(v\check{v}\) to be the number \(c\) of horizontal (alternatively, vertical) arcs of \(D\) above \(\partial\).

Proof. If addition and subtraction in \([n]\) are taken modulo \(n\) and we write \([y,x)=\{y,y+1,y+2,\ldots,x-1\}\), for \(x,y\in[n]\), and \(S^c=[n]\setminus S\), for \(S=\{i\in[n]:b_i=1\}\subseteq[n]\), then the cardinalities of the sets \(\{y\in S^c\setminus x: |[y,x)\cap S|< |[y,x)\cap S^c|\}\) yield all the edge colors, where \(x\in S^c\) varies.

The Lexical Procedure of Theorem 8 yields a 1-fac\-tor\-i\-za\-tion not only for \(M_k/\pi\) but also for \(R_k\) and \(M_k\). This is clarified by the end of Section 8.

8. Uncastling and lexical 1-factorization

A notation \(\delta(v)\) is assigned to each pair \(\{v,\aleph_\pi(v)\}\in R_k\), where \(v\in L_k/\pi\), so that there is a unique \(k\)-germ \(\alpha=\alpha(v)\) with \(\langle F(\alpha)\rangle=\delta(v)\), (where the notation \(\langle\cdot\rangle\) is as in \(\langle B_A\rangle\) in Section 6). We exemplify \(\delta(v)\) for \(k=2\) in Figure 2, with the Lexical Procedure (indicated by arrows ``\(\Rightarrow\)'') departing from \(v=(00011)\) (top) and \(v=(00101)\) (bottom), passing to sketches of \(\Gamma\) (separated by symbols ``+''), one sketch (in which to trace the edges of \(D\subset\Gamma\) as in Theorem 8) per representative \(b_0b_1\cdots b_{n-1}=0b_1\cdots b_{n-1}\) of \(v\) shown under the sketch (where \(b_0=0\) is underscored) and pointing via an arrow ``\(\rightarrow\)'' to the corresponding color \(c\in[k+1]\). Recall this \(c\) is the number of horizontal arcs of \(D\) below \(\partial\).
In each of the 2 cases in Figure 2 (top, bottom), an arrow ``\(\Rightarrow\)'' to the right of the sketches points to a modification \(\hat{v}\) of \(b_0b_1\cdots b_{n-1}=0b_1\cdots b_{n-1}\) obtained by setting as a subindex of each 0 (resp. 1) its associated color \(c\) (resp. an asterisk ``\(*\)'' ). Further to the right, a third arrow ``\(\Rightarrow\)'' points to the \(n\)-tuple \(\delta(v)\) formed by the string of subindexes of entries of \(\hat{v}\) in the order they appear from left to right.

Theorem 6. Let \(\alpha(v^0)=a_{k-1}\cdots a_1=00\cdots 0\). To each \(\delta(v)\) corresponds a sole \(k\)-germ \(\alpha=\alpha(v)\) with \(\langle F(\alpha)\rangle=\delta(v)\) by means of the following ``Uncastling Procedure'': Given \(v\in L_k/\pi\), let \(W^i=01\cdots i\) be the maximal initial numeric (i.e., colored) substring of \(\delta(v)\), so the length of \(W^i\) is \(i+1\) (\(0\le i\le k\)). If \(i=k\), let \(\alpha(v)=\alpha(v^0)\); else, set \(m=0\) and:

• set \(\delta(v^m)=\langle W^i|X|Y|Z^i\rangle\), where \( Z^i\) is the terminal \(j_m\)-substring of \(\delta(v^m)\), with \(j_m=\) \(i+1\), and let \(X,Y\) (in that order) start at contiguous numbers \(\Omega\) and \(\Omega-1\ge i\);
• set \(\delta(v^{m+1})=\langle W^i|Y|X|Z^i\rangle\);
• obtain \(\alpha(v^{m+1})\) from \(\alpha(v^m)\) by increasing its entry \(a_{j_m}\) by 1;
• if \(\delta(v^{m+1})=[01\cdots k*\cdots*]\), then stop; else, increase \(m\) by 1 and go to step 1.

Proof. This is a procedure inverse to that of Castling (Section 3), so 1-4 follow.

Theorem~\ref{un} allows to produce a finite sequence \(\delta(v^0),\delta(v^1),\ldots,\delta(v^m),\ldots,\) \(\delta(v^s)\) of \(n\)-strings with \(j_0\ge j_1\ge\cdots\ge j_m\cdots\ge j_{s-1}\) as in steps 1-4, and \(k\)-germs \(\alpha(v^0),\alpha(v^1),\ldots,\) \(\alpha(v^m),\ldots,\) \(\alpha(v^s)\), taking from \(\alpha(v^0)\) through the \(k\)-germs \(\alpha(v^m)\), (\(m=1,\ldots,s-1\)), up to \(\alpha(v)=\alpha(v^s)\) via unit incrementation of \(a_{j_m}\), for \(0\le m< s\), where each incrementation yields the corresponding \(\alpha(v^{m+1})\). Recall \(F\) is a bijection from the set \(V({\mathcal T}_k)\) of \(k\)-germs onto \(V(R_k)\), both sets being of cardinality \(C_k\). Thus, to deal with \(V(R_k)\) it is enough to deal with \(V({\mathcal T}_k)\), a fact useful in interpreting Theorem~\ref{thm4}. For example \(\delta(v^0)=\langle 04*3*2*1\,*\rangle=\langle 0|4*|3*2*1|*\rangle=\langle W^0|X|Y|Z^0\rangle\) with \(m=0\) and \(\alpha(v^0)=123\), continued in Table \ref{tab3} with \(\delta(v^1)=\langle W^0|Y|X|Z^0\rangle\), finally arriving to \(\alpha(v^s)=\alpha(v^6)=000\).

A pair of skew edges \((B_A)\aleph_\pi((B_{A'}))\) and \((B_{A'})\aleph((B_A))\) in \(M_k/\pi\), to be called a skew reflection edge pair ( SREP), provides a color notation for any \(v\in L_{k+1}/\pi\) such that in each particular edge class mod \(\pi\):

• (I) all edges receive a common color in \([k+1]\) regardless of the endpoint on which the Lexical Procedure (or its modification immediately below) for \(v\in L_{k+1}/\pi\) is applied;
• (II) the 2 edges in each SREP in \(M_k/\pi\) are assigned a common color in \([k+1]\).

The modification in step (I) consists in replacing in Figure 2 each \(v\) by \(\aleph_\pi(v)\) so that on the left we have instead now \((00111)\) (top) and \((01011)\) (bottom) with respective sketch subtitles $$\begin{array}{ccc} ^{0011\underline{1}\rightarrow 0,}_{0101\underline{1}\rightarrow 0,}& ^{1001\underline{1}\rightarrow 1,}_{1010\underline{1}\rightarrow 2,}& ^{1100\underline{1}\rightarrow 2,}_{0110\underline{1}\rightarrow 1,}\\ \end{array}$$ resulting in similar sketches when the steps (1)-(5) of the Lexical Procedure are taken with right-to-left reading and processing of the entries on the left side of the subtitles (before the arrows ``\(\rightarrow\)''), where the values of each \(b_i\) must be taken complemented, (i.e., as \(\bar{b_i}\)).
Since an SREP in \(M_k\) determines a unique edge \(\epsilon\) of \(R_k\) (and vice versa), the color received by the SREP can be attributed to \(\epsilon\), too. Clearly, each vertex of either \(M_k\) or \(M_k/\pi\) or \(R_k\) defines a bijection from its incident edges onto the color set \([k+1]\). The edges obtained via \(\aleph\) or \(\aleph_\pi\) from these edges have the same corresponding colors.

Theorem 7. A 1-factorization of \(M_k/\pi\) by the colors \(0,1,\ldots,k\) is obtained via the Lexical Procedure that can be lifted to a covering 1-factorization of \(M_k\) and subsequently collapsed onto a folding 1-factorization of \(R_k\). This insures the notation \(\delta(v)\) for each \(v\in V(R_k)\) so that there is a unique \(k\)-germ \(\alpha=\alpha(v)\) with \(\langle F(\alpha)\rangle=\delta(v)\).

Proof. As pointed out in (II) above, each SREP in \(M_k/\pi\) has its edges with a common color in \([k+1]\). Thus, the \([k+1]\)-coloring of \(M_k/\pi\) induces a well-defined \([k+1]\)-coloring of \(R_k\). This yields the claimed collapsing to a folding 1-factorization of \(R_k\). The lifting to a covering 1-factorization in \(M_k\) is immediate. The arguments above determine that the collapsing 1-factorization in \(R_k\) induces the claimed \(k\)-germs \(\alpha(v)\).

9. Union of lexical 1-factors of colors 0 and 1

Given a \(k\)-germ \(\alpha\), let \((\alpha)\) represent the dihedral class \(\delta(v)=\langle F(\alpha)\rangle\) with \(v\in L_k/\pi\). Let \(W_{01}^k\) be the 2-factor given by the union of the 1-factors of colors 0 and 1 in \(M_k\) (namely those formed by lifting the edges \(\alpha\alpha^0\) and \(\alpha\alpha^1\) of \(R_k\) in the notation of \ Section 12, instead of those of colors \(k\) and \(k-1\), as in [7]). The cycles of \(W_{01}^k\) are grown in this section from specific paths, as suggested in Figure 3 for \(k=2,3,4\) (say: cycle \(C_0\) that starts with vertically expressed path \(X(0)\), for \(k=2\); cycles \(C_0,C_1\) that start with vertically expressed paths \(X(0),X(1)\), for \(k=3\); and cycles \(C_0,C_1,C_2\) that start with vertically expressed paths \(X(0),X(1),X(2)\), for \(k=4\)). Here, vertices \(v\in L_k\) (resp. \(v\in L_{k+1}\)) will be represented with:

• (a) 0- (resp. 1-) entries replaced by their respective colors \(0,1,\ldots,k\) (resp. asterisks);
• (b) 1- (resp. 0-) entries replaced by asterisks (resp. their respective colors \(0,1,\ldots,k\));
• (c) delimiting chevron symbols "\(>\)" or ``\(\rangle\)'' (resp. "\(< \)" or ``\(\langle\)''), instead of parentheses or brackets, indicating the reading direction of ``forward'' (resp. ``backward'') \(n\)-tuples.

Each such vertex \(v\) is showen to belong (via the set membership symbol expressed by ``\(\varepsilon\)'' in Figure 3) to \((\alpha_v)\), where \(\alpha_v\) is the \(k\)-germ of \(v\), also expressed as its (underlined decimal) natural order. In each case, Figure 3 shows a vertically presented path \(X(i)\) of length \(4k-1\) in the corresponding cycle \(C_i\) starting at the vertex \(w=b_0b_1\cdots b_{2k}=01\cdots *\) of smallest natural order and proceeding by traversing the edges colored 1 and 0, alternatively. The terminal vertex of such subpath is \(b_{2k}b_0b_1\cdots b_{2k-1}=*01\cdots b_{2k-1}\), obtained by translation mod \(n\) from \(w\).

Observation 1. The initial entries of the vertices in each \(C_i\) are presented downward, first in the 0-column of \(X(i)\), then in the \((2k-j)\)-column of \(X(i)\), (\(j\in[2k]\), only up to \(|C_i|\).

In Figure 3, initial entries are red if they are in \(\{0,1\}\) and each cycle \(C_i\) is encoded on its top right by a vertical sequence of expressions \((0\cdots 1)(1\cdots 0)\) that allows to get the sequence of initial entries of the succeeding vertices of \(C_i\) by interspersing asterisks between each 2 terms inside parentheses (), then removing those ().
An ordered tree \(T=T_v\) (Remark 1) for each \(v\) in the exemplified \(C_i(=C\) in Proposition 2(v) [7]) is shown at the lower right of its case in Figure 3. Each of these \(T_v\) for a specific \(C_i\) corresponds to the \(k\)-germ \(\alpha_v\) (so we write \(F(\alpha_v)=F(T_{\alpha_v})\)) and is headed in the figure by its (underlined decimal) natural order. In the figure, vertices of each \(T_v\) are denoted \(i\), instead of \(v_i\) (\(i\in[k+1]\)). The trees corresponding to the \(k\)-germs in each case are obtained by applying root rotation [7], consisting in replacing the tree root by its leftmost child and redrawing the ordered tree accordingly.
A plane tree is an equivalence relation of ordered trees under root rotations. In the notation of Section 12, applying one root rotation has the same effect as traversing first an edge \(\alpha\alpha_0\) in \(C_i\) and then edge \(\beta\beta_1\), also in \(C_i\), where \(\beta=\alpha_0\). In each case, a yellow box shows a subpath of \(X(i)\) with \(\frac{1}{n}|V(C_i)|\) vertices of \(C_i\) that takes into account the rotation symmetry of the associated plane tree \({\mathcal T}_i\).
In Figure 3 for \(k=4\), successive application of root rotations on the second cycle, \(C_1\), produces the cycle \((\underline{9},\underline{2},\underline{4},\underline{11},\underline{5},\underline{6},\underline{12},\underline{7})\), the square graph of \(C_1\), that starts downward from the second row or upward from the third row, thus covering respectively the vertices of \(L_4\) or \(L_5\) in the class.
Let \(D_i\) (\(i\ge 0\)) be the set of substrings of length \(2i\) in an \(F(\alpha)\) with exactly \(i\) color-entries such that in every prefix (i.e. initial substring), the number of asterisk-entries is at least as large as the number of color-entries. The elements of \(D=\cup_{i\ge 0}D_i\) are known as Dyck words. Each \(F(\alpha)\) is of the form \(0v1u*\), where \(u\) and \(v\) are Dyck words and 0 and 1 are colors in \([k+1]\) [7]. The ``forward'' \(n\)-tuple \(F(\alpha)\) in \((\underline{9},\underline{2},\underline{4},\underline{11},\underline{5},\underline{6},\underline{12},\underline{7})\) can be written with parentheses enclosing such Dyck words \(v\) and \(u\), namely \(0()1(34**2*)*\), for \(\underline{2}\); \(0(3*24**)1()*\), for \(\underline{9}\); \(0()1(3*24**)*\), for \(\underline{7}\); \(0(3*2*)1(4*)*\), for \(\underline{12}\); \(0(24*3**)1()*\), for \(\underline{6}\); \(0()1(24*3**)*\), for \(\underline{5}\); \(0(2*)1(4*3*)*\), for \(\underline{11}\); and \(0(34**2*)1()*\), for \(\underline{4}\). Similar treatment holds for ``backward'' \(n\)-tuples.
As in Figure 3, each pair \((0\cdots 1)(1\cdots 0)\) represents 2 paths in the corresponding cycle, of lengths \(2|(0\cdots 1)|-1\) and \(2|(1\cdots 0)|\),\ adding up to \(4k+2\). If these 2 paths are of the form \(0v1u*\) and \(0v'1u'*\) (this one read in reverse), then \(|u|+2=|(0\cdots 1)|\) and \(|u'|+2=|(1\cdots 0)|\). Reading these paths starts at a 0-entry and ends at a 1-entry. In reality, the collections of paths obtained from the 1-factors here have the leftmost entry of the \(n\)-tuples representing their vertices constantly equal to \(1\in\mathbb{Z}_2\) before taking into consideration (items (a) and (b) above, but with the reading orientations given in item (c).
The 1-factor of color 0 makes the endvertices of each of its edges to have their representative plane trees obtained from each other by horizontal reflection \(\Phi=F\alpha^0F^{-1}\). For example, Figure 3 shows that for \(k=2\): both \(\underline{0},\underline{1}\) in \(X(0)\) are fixed via \(\Phi\); for \(k=3\): \(\underline{0}\) in \(X(0)\) is fixed via \(\Phi\) and \(\underline{1},\underline{3}\) in \(X(0)\) correspond to each other via \(\Phi\); and \(\underline{2},\underline{4}\) in \(X(1)\) are fixed via \(\Phi\); for \(k=4\): \(\underline{0},\underline{8}\) in \(X(0)\) are fixed via \(\Phi\) and \(\underline{1},\underline{3}\) in \(X(0)\) correspond to each other via \(\Phi\); \(\underline{5},\underline{9}\) in \(X(1)\) are fixed via \(\Phi\) and the pairs \((\underline{5},\underline{9})\), \((\underline{2},\underline{7})\), \((\underline{4},\underline{12})\) and \((\underline{6},\underline{11})\) in \(X(1)\) are pairs of correspondent plane trees via \(\Phi\); and \(\underline{10},\underline{13}\) in \(X(2)\) are fixed via \(\Phi\). This horizontal reflection symmetry arises from Theorem 3. It accounts for each pair of contiguous rows in any \(X(i)\) corresponding to a 0-colored edge.
For \(k=5\), this symmetry via \(\Phi\) occurs in all cycles \(C_i\) (\(i\in[6]\)). But we also have \(F(\underline{22})=\rangle 024**135***\rangle\), where \(\underline{22}=(1111)\) in \(C_0\) and \(F(\underline{39})=\rangle 03*2*15*4**\rangle\), where \(\underline{39}=(1232)\) in \(C_3\), both having their 1-colored edges leading to reversed reading between \(L_5\) and \(L_6\), again by Theorem 3. Moreover, \(F((11\cdots 1))\) has a similar property only if \(k\) is odd; but if \(k\) is even, a 0-colored edge takes place, instead of the 1-colored edge for \(k\) odd. These cases reflect the following lemma (which can alternatively be implied from Theorem 10 (B)-(C)) via the correspondence \(i\leftrightarrow k-i\), (\(i\in[k+1]\)).

Observation 2.

• (A) Every \(0\)-colored edge is represented via \(\Phi=F\alpha^0F^{-1}\).
• (B) Every 1-colored edge is represented via the composition \(\Psi\) of \(\Phi\) (first) and root rotation (second).

By Theorem 3(ii), the number \(\xi\) of contiguous pairs of vertices of \(M_k\) in each \(C_i\) with a common \(k\)-germ happens in pairs. The first cases for which this \(\xi\) is null happens for \(k=6\), namely for the 2 reflection pairs \((\rangle 012356**4****\rangle,\rangle 01235*46*****\rangle)\) and \((\rangle 01246*5**3***\rangle,\rangle 0124*36*5****\rangle)\) whose respective ordered trees are enantiomorphic, i.e they are reflection via \(\Phi\) of each other. We say that these 2 cases are enantiomorphic. In fact, the presence of a pair of enantiomorphic ordered trees in a case of an \(X(i)\) will be distinguished by saying that the case is enantiomorphic. For example, \(k=4\) offers \((\underline{1},\underline{3})\) as the sole enantiomorphic pair in \(X(0)\), and \((\underline{2},\underline{7})\), \((\underline{4},\underline{12})\), \((\underline{11},\underline{6})\) as all the enantiomorphic pairs in \(X(1)\). Each enantiomorphic cycle \(C_i\) or each cycle \(C_i\) with \(\xi=2\) has \(|C_i|=2k(4k+2)\). If \(\xi=2\zeta\) with \(\zeta>1\), then \(|C_i|=\frac{2k(4k+1)}{\zeta}\). On account of these facts, we have the following:

Observation 3. For each integer \(k>1\), there is a natural bijection \(\Lambda\) from the \(k\)-edge plane trees onto the cycles of \(W_{01}^k\), as well as a partition \({\mathcal P}_k\) of the \(k\)-germs (or the ordered trees they represent via \(F\)), with each class of \({\mathcal P}_k\) in natural correspondence either to a \(k\)-edge plane tree or to a pair of enantiomorphic \(k\)-edge plane trees disconnecting in \(W_{01}^k\) the forward (in \(L_k\)) and reversed (in \(L_{k+1}\)) readings of each vertex \(v\) of their associated cycles via \(\Lambda\).

10. Reinterpretation of the middle-levels theorem

For each cycle \(C_i\) of \(W_{01}^k\), the ordered trees of its plane tree \({\mathcal T}_i\) with leftmost subpath of length 1 from \(v_0\) to a vertex \(v_h\) determine 6-cycles touching \(C_i\) in 2 nonadjacent edges as follows:
Let \(t_i< k\) be the number of degree 1 vertices of \({\mathcal T}_i\). Let \(\tau_i\) be the number of rotation symmetries of \({\mathcal T}_i\). Then, there are \(\frac{t_i}{\tau_i}\) classes mod \(n\) of pairs of vertices \(u,v\) at distance 5 in \(C_i\) with \(u\in L_{k+1}\) ahead of \(v\in L_k\) in \(X(i)\) and associated color \(h\in\{2,\ldots,k\}\) such that:

• (i) \(u\) and \(v\) are adjacent via \(h\);
• (ii) \(u\) (resp. \(v\)) has cyclic backward (resp. forward) reading \(\langle\cdot\cdots * h 0 * \cdots\langle\) (resp. \(\rangle\cdots * 0 h *\cdots\cdot\rangle\));
• (iii) the column in which the occurrences of \(h\) in (ii) happen at distance 5 looks between \(u\) and \(v\) (both included) as the transpose of \((h,*,0,0,*,h)\). Recall there are \(n\) such columns.

In each case, the vertices \(u'\) and \(v'\) in \(C_i\) preceding respectively \(u\) and \(v\) in \(X_i\) are endvertices of a 3-path \(u'u''v''v'\) in \(M_k\) with the edge \(u''v''\) in a cycle \(C_j\ne C_i\) in \(W_{01}^k\).
The 6-cycle \(U_i^j=(uu'u''v''v'v)\) has as symmetric difference \(U_i^j\Delta(C_i\cup C_j)\) a cycle in \(M_k\) whose vertex set is \(V(C_i\cup C_j)\). With \(u',u'',v'',v',v,u,u'\) shown vertically, Figure 4 illustrates \(U_i^j\), twice each for \(k=3,4\).
That symmetric difference replaces respectively the edges \(u''v''\), \(v'v\), \(uu'\) in \(C_i\cup C_j\) by \(u'u''\), \(v''v''\), \(vu\). In the figure, vertically contiguous positions holding a common number \(g\) (meaning adjacency via color \(g\)) are presented in red if \(g\in\{0,1\}\), e.g. \(u''v''\) with \(g=1\), in column say \(r_1\) exactly at the position where \(u\) and \(v\) differ, (however having common color \(h\) as in (i) above) and in orange, otherwise. The column \(r_2\) (resp \(r_3\)) in each instance of Figure 3 containing color 1 in \(u'\) (resp. 0 in \(v'\)) and a color \(c\in\{2,\ldots,k\}\) in \(v'\) (resp. color \(d\in\{2,\ldots,k\}\) in \(u'\)) starts with \(1,*,c,c\), (resp. \(d,d,*,0\)), where \(c,d\in\{2,\ldots,k\}\). Then, \(r_1,r_2,r_3\) are the only columns having changes in the binary version of \(U_i^j\). All other columns have their first 4 entries alternating asterisks and colors. In the first disposition in item (ii) above, we have that: {\bf(ii')} \(u''\) (resp. \(v''\)) has cyclic backward (resp. forward) reading \(\langle\cdot\cdots d 1 0 * \cdots\langle\) (resp. \(\rangle\cdot\cdots * 1 * 0\cdots\rangle\)).
In the previous paragraph, ``ahead'' can be replaced by ``behind'', yielding additional 6-cycles \(U_i^j\) by modifying adequately the accompanying text.

Theorem 8.[6, 7] Let \(0< k\in\mathbb{Z}\). A Hamilton cycle in \(M_k\) is obtained by means of the symmetric differences of \(W_{01}^k\) with the members of a set of pairwise edge-disjoint 6-cycles \(U_i^j\).

Proof. Clearly, the statement holds for \(k=1\). Assume \(k>1\). Let \(\mathcal D\) be the digraph whose vertices are the cycles \(C_i\) of \(W_{01}^k\), with an arc from \(C_i\) to \(C_j\), for each 6-cycle \(U_i^j\), where \(C_i,C_j\in V(\mathcal D)\) with \(i\ne j\). Since \(M_k\) is connected, then \(\mathcal D\) is connected.
Moreover, the outdegree and indegree of every \(C_i\) in \(\mathcal D\) is \(2n\frac{t_i}{\tau_i}\) (see items (ii) and (ii') above), in proportion with the length of \(C_i\), a \(\frac{1}{n}\)-th of which is illustrated in each yellow box of Figure 3.
Consider a spanning tree \({\mathcal D}'\) of \(\mathcal D\). Since all vertices of \(\mathcal D\) have outdegree \(>0\), there is a \({\mathcal D}'\) in which the outdegree of each vertex is 1. This way, we avoid any pair of 6-cycles \(U_i^j\) with common \(C_i\) in which the associated distance-6 subpaths from \(u'\) to \(v\) in \(C_i\) do not have edges in common. For each \(a\in A({\mathcal D}')\), let \(\nabla(a)\) be its associated 6-cycle \(U_i^j\). Then, \(\{\nabla(a);a\in A({\mathcal D}')\}\) can be selected as a collection of edge-disjoint 6-cycles. By performing all symmetric differences \(\nabla(a)\Delta(C_i\cup C_j)\) corresponding to these 6-cycles, a Hamilton cycle is obtained.

11. Alternate viewpoint on ordered trees

To have the viewpoint of [7], replace \(v_0\) by \(v_k\) as root of the ordered trees. We start with examples. Figure 5 shows on its left-hand side the 14 ordered trees for \(k=4\) encoded at the bottom of Table 1. Each such tree \(T=T(\alpha)\) is headed on top by its \(k\)-germ \(\alpha\), in which the entry \(i\) producing \(T\) via Castling is in red. Such \(T\) has its vertices denoted on their left and its edges denoted on their right, with their notation \(v_i\) and \(e_j\) given in Section 9. Castling here is indicated in any particular tree \(T=T(\alpha)\ne T(00\cdots 0)\) by distinguishing in red the largest subtree common with that of the parent tree of \(T\) (as in Theorem 1) whose Castling reattachment produces \(T\). This subtree corresponds with substring \(X\) in Theorem 2. In each case of such parent tree, the vertex in which the corresponding tree-surgery transformation leads to such a child tree \(T\) is additionally labeled (on its right) with the expression of its \(k\)-germ, in which the entry to be modified in the case is set in red color.
On the other hand, the 14 trees in the right-hand side of Figure 5 have their labels set by making the root to be \(v_k\) (instead of \(v_0\)), then going downward to \(v_0\) (instead of \(v_k\)) while gradually increasing (instead of decreasing) a unit in the subindex \(j\) of the denomination \(v_j\), sibling by sibling from left to right at each level. The associated \(k\)-germ headers on this right-hand side of the figure correspond to the new root viewpoint. This determines a bijection \(\Theta\) established by correspondence between the old and the new header \(k\)-germs. In our example, it yields an involution formed by the pairs \((001,100)\), \((011,110)\), \((120,012)\) and \((112,121)\), with fixed 000, 010, 101, 111, 122 and 123.

The function \(\Theta\) seen from the \(k\)-germ viewpoint, namely as the composition function \(F^{-1}\Theta F\), behaves as follows. Let \(\alpha=a_{k-1}a_{k-2}\cdots a_2a_1\) be a \(k\)-germ and let \(a_i\) be the rightmost occurrence of its largest integer value. A substring \(\beta\) of \(\alpha\) is said to be an atom if it is either formed by a sole 0 or is a maximal strictly increasing substring of \(\alpha\) not starting with 0. For example, consider the 17-germ \(\alpha'=0123223442310121\). By enclosing the successive atoms between parentheses, \(\alpha'\) can be written as \(\alpha'=(0)123(2)(234)4(23)(1)(0)(12)(1)\), obtained by inserting in a base string \(\gamma'=1\cdots a_i=1234\) all those atoms according to their order, where \(\gamma'\) appears partitioned into subsequent un-parenthesized atoms distributed and interspered from left to right in \(\alpha'\) just once each as further to the right as possible.
This atom-parenthesizing procedure works for every \(k\)-germ \(\alpha\) and determines a corresponding base string \(\gamma\), like the \(\gamma'\) in our example.

Theorem 9. Given a \(k\)-germ \(\alpha=a_{k-1}a_{k-2}\cdots a_2a_1\), let \(a_i\) be the leftmost occurrence of its largest integer value. Then, \(\alpha\) is obtained from a base string \(\gamma=1\cdots a_i\) by inserting in \(\gamma\) all atoms of \(\alpha\setminus \gamma\) in their left-to-right order. Moreover, \(F^{-1}\Theta F(\alpha)\) is obtained by reversing the insertion of those atoms in \(\gamma\), in right-to-left fashion. For both insertions, \(\gamma\) is partitioned into subsequent atoms distributed in \(\alpha\) and \(F^{-1}\Theta F(\alpha)\) as further to the right as possible.

Proof. \(F^{-1}\Theta F(\alpha)\) is obtained by reversing the position of the parenthesized atoms, inserting them between the substrings of a partition of \(\gamma\) as the one above but for this reversing situation. In the above example of \(\alpha'\), it is \(F^{-1}\Theta F(\alpha')=(1)(12)(0)(1)12(234)34(23)(2)(0)\).

12. Germ structure of 1-factorizations

We present each vertex of \(R_k\) via the pair \(\delta(v)=\{v,\aleph_\pi(v)\}\in V(R_k)\) (\(v\in L_k/\pi\)) of Section 8 and via the \(k\)-germ \(\alpha\) for which \(\delta(v)=\langle F(\alpha)\rangle\), and view \(R_k\) as the graph whose vertices are the \(k\)-germs \(\alpha\), with adjacency inherited from that of their \(\delta\)-notation via \(F^{-1}\) (i.e. Uncastling). So, \(V(R_k)\) is presented as in the natural (\(k\)-germ) enumeration (see Section 2 and Subsection 1.3 [8]).
To start with, examples of such presentation are shown in Table 4 for \(k=2\) and \(3\), where \(m\), \(\alpha=\alpha(m)\) and \(F(\alpha)\) are shown in the first 3 columns, for \(0\le m< C_k\). The neighbors of \(F(\alpha)\) are presented in the central columns of the table as \(F^k(\alpha)\), \(F^{k-1}(\alpha)\), \(\ldots\), \(F^0(\alpha)\) respectively for the edge colors \(k,k-1,\ldots,0\), with notation given via the effect of function \(\aleph\). The last columns yield the \(k\)-germs \(\alpha^k\), \(\alpha^{k-1}\), \(\ldots\), \(\alpha^0\) associated via \(F^{-1}\) respectively to the listed neighbors \(F^k(\alpha)\), \(F^{k-1}(\alpha)\) , \(\ldots\), \(F^0(\alpha)\) of \(F(\alpha)\) in \(R_k\).

For \(k=4\) and 5, Tables 5 and 6 have a similar respective natural enumeration adjacency disposition. We can generalize these tables directly to Colored Adjacency Tables denoted CAT\((k)\), for \(k>1\). This way, Theorem 10(A) below is obtained as indicated in the aggregated last row upending Tables 5 and 6 citing the only non-asterisk entry, for each of \(i=k,k-2,\ldots,0\), as a number \(j=(k-1),\ldots,1\) that leads to entry equality in both columns \(\alpha=a_{k-1}\cdots a_j\cdots a_1\) and \(\alpha^i=a_{k-1}^i\cdots a_j^i\cdots a_1^i\), that is \(a_j=a_j^i\). Other important properties are contained in the remaining items of Theorem 10, including (B), that the columns \(\alpha^0\) in all CAT\((k)\), (\(k>1\)), yield an (infinte) integer sequence.

Theorem 10. Let: \(k>1\), \(j(\alpha^k)=k-1\) and \(j(\alpha^{i-1})=i\), (\(i=k-1,\ldots,1\)). Then:

• (A) each column \(\alpha^{i-1}\) in CAT\((k)\), for \(i\in[k]\cup\{k+1\}\), preserves the respective \(j(\alpha^{i-1})\)-th entry of \(\alpha\);
• (B) the columns \(\alpha^k\) of all CAT\((k)\)'s for \(k>1\) coincide into an RGS sequence and thus into an integer sequence \({\mathcal S}_0\), the first \(C_k\) terms of which form an idempotent permutation for each \(k\);
• (C) the integer sequence \({\mathcal S}_1\) given by concatenating the \(m\)-indexed intervals \([0,2),[2,5), \ldots,\) \([C_{k-1},C_k)\), etc. in column \(\alpha^{k-1}\) of the corresponding tables CAT\((2)\), CAT\((3),\ldots\), CAT\((k)\), etc. allows to encode all columns \(\alpha^{k-1}\)'s;
• (D) for each \(k>1\), there is an idempotent permutation given in the \(m\)-indexed interval \([0,C_k)\) of the column \(\alpha^{k-1}\) of CAT\((k)\); such permutation equals the one given in the interval \([0,C_k)\) of the column \(\alpha^{k-2}\) of CAT\((k+1)\).

Proof. (A)holds as a continuation of the observation made above with respect to the last aggregated row in Figure 5. Let \(\alpha\) be a \(k\)-germ. Then \(\alpha\) shares with \(\alpha^k\) (e.g. the leftmost column \(\alpha^i\) in Tables 4, 5 and 6, for \(0\le i\le k\)) all the entries to the left of the leftmost entry 1, which yields (B). Note that if \(k=3\) then \(m=2,3,4\) yield for \(\alpha^{k-1}\) the idempotent permutation \((2,0)(4,1)\), illustrating (C). (D) can be proved similarly.

The sequences in Theorem 10 (B)-(C) start as follows, with intervals ended in ``;'': $$\begin{array}{rrrrrrrrrrrrrrrrrr} ^{\{0\}\cup\mathbb{Z}^+=}&^{0,}&^{1;}&^{2,}&^{3,}&^{4;}&^{5,}&^{6,}&^{7,}&^{8,}&^{9,}&^{10,}&^{11,}&^{12,}&^{13;}&^{14}&^{15,}&^{16,\ldots}\\\hline ^{(B)=}_{(C)=}&^{0,}_{1,}&^{1;}_{0;}&^{3,}_{0,}&^{2,}_{3,}&^{4;}_{1;}&^{7,}_{0,}&^{9,}_{1,}&^{5,}_{8,}&^{8,}_{7,}&^{\hspace{1mm}6,}_{12,}&^{12,}_{\hspace{1mm}3,}&^{11,}_{\hspace{1mm}2,}&^{10,}_{\hspace{1mm}9,}&^{13;}_{\hspace{1mm}4;}&^{19,}_{\hspace{1mm}0,}&^{20,}_{\hspace{1mm}1,}&^{25,\ldots}_{\hspace{1mm}3,\ldots}\\ \end{array}$$

Remark 2. With the notation of Section 11 and Theorem 9, for each of the involutions \(\alpha^i\) (\(0< i< k \)), it holds that \(\alpha^i\Theta=\Theta\alpha^{k-i}\). This implies that

• (A) every \(k\)-colored edge represents an adjacency via \(\Phi'=F\alpha_kF^{-1}\) and
• (B) every \((k-1)\)-colored edge represents an adjacency via \(\Psi'=F\Psi F^{-1}\).

In addition, the reflection symmetry of \(\Phi'\) yields the sequence \(S_0\) cited in Theorem 10(B). A similar observation yields from \(\Psi'\) the sequence \(S_1\) cited in Theorem 10(C).

Given a \(k\)-germ \(\alpha=a_{k-1}\cdots a_1\), we want to express \(\alpha^k,\alpha^{k-1},\ldots,\alpha^0\) as functions of \(\alpha\). Given a substring \(\alpha'=a_{k-j}\cdots a_{k-i}\) of \(\alpha\) (\(0< j\le i< k\)), let:

• (a) the reverse string off \(\alpha'\) be \(\psi(\alpha')=a_{k-i}\cdots a_{k-j}\);
• (b) the ascent of \(\alpha'\) be
  • (i) its maximal initial ascending substring, if \(a_{k-j}=0\), and
  • (ii) its maximal initial non-descending substring with at most 2 equal nonzero terms, if \(a_{k-j}>0\).

Then, the following remarks allow to express the \(k\)-germs \(\alpha^p=\beta=b_{k-1}\cdots b_1\) via the colors \(p=k,k-1,\ldots,0\), independently of \(F^{-1}\) and \(F\).

Remark 3. Assume \(p=k\). If \(a_{k-1}=1\), take \(0|\alpha\) instead of \(\alpha=a_{k-1}\cdots a_1\), with \(k-1\) instead of \(k\), removing afterwards from the resulting \(\beta\) the added leftmost 0. Now, let \(\alpha_1=a_{k-1}\cdots a_{k-i_1}\) be the ascent of \(\alpha\). Let \(B_1=i_1-1\), where \(i_1=||\alpha_1||\) is the length of \(\alpha_1\). It can be seen that \(\beta\) has ascent \(\beta_1=b_{k-1}\cdots b_{k-i_1}\) with \(\alpha_1+\psi(\beta_1)=B_1\cdots B_1\). If \(\alpha\ne\alpha_1\), let \(\alpha_2\) be the ascent of \(\alpha\setminus\alpha_1\). Then there is a \(||\alpha_2||\)-germ \(\beta_2\) with \(\alpha_2+\psi(\beta_2)=B_2\cdots B_2\) and \(B_2=||\alpha_1||+||\alpha_2||-2\). Inductively when feasible for \(j>2\), let \(\alpha_j\) be the ascent of \(\alpha\setminus(\alpha_1|\alpha_2|\cdots|\alpha_{j-1})\). Then there is a \(||\alpha_j||\)-germ \(\beta_j\) with \(\alpha_j+\psi(\beta_j)=B_j\cdots B_j\) and \(B_j=||\alpha_{j-1}||+||\alpha_j||-2\). This way, \(\beta=\beta_1|\beta_2|\cdots|\beta_j|\cdots\).

Remark 4. Assume \(k>p>0\). By Theorem 10 (A), if \(p< k -1\), then \(b_{p+1}=a_{p+1}\); in this case, let \(\alpha'=\alpha\setminus\{a_{k-1}\cdots a_q\}\) with \(q=p+1\). If \(p=k-1\), let \(q=k\) and let \(\alpha'=\alpha\). In both cases (either \(p< k-1\) or \(p=k-1\)) let \(\alpha'_1=a_{q-1}\cdots a_{k-i_1}\) be the ascent of \(\alpha'\). It can be seen that \(\beta'=\beta\setminus\{b_{k-1}\cdots b_q\}\) has ascent \(\beta'_1=b_{k-1}\cdots b_{k-i_1}\) where \(\alpha'_1+\psi(\beta'_1)=B'_1\cdots B'_1\) with \(B'_1=i_1+a_q\). If \(\alpha'\ne\alpha'_1\) then let \(\alpha'_2\) be the ascent of \(\alpha'\setminus\alpha'_1\). Then there is a \(||\alpha'_2||\)-germ \(\beta'_2\) where \(\alpha'_2+\psi(\beta'_2)=B'_2\cdots B'_2\) with \(B'_2=||\alpha'_1||+||\alpha'_2||-2\). Inductively when feasible for \(j>2\), let \(\alpha_j\) be the ascent of \(\alpha'\setminus(\alpha'_1|\alpha'_2|\cdots|\alpha'_{j-1})\). Then there is a \(||\alpha'_j||\)-germ \(\beta'_j\) where \(\alpha'_j+\psi(\beta'_j)=B'_j\cdots B'_j\) with \(B'_j=||\alpha'_{j-1}||+||\alpha'_j||-2\). This way, \(\beta'=\beta'_1|\beta'_2|\cdots|\beta'_j|\cdots\).
We process the left-hand side from position \(q\). If \(p>1\), we set \(a_{a_q+2}\cdots a_q +\psi(b_{b_q+2}\cdots b_q)\) to equal a constant string \(B\cdots B\), where \(a_{a_q+2}\cdots a_q\) is an ascent and \(a_{a_q+2}=b_{b_q+2}\). Expressing all those numbers \(a_i,b_i\) as \(a_i^0,b_i^0\), respectively, in order to keep an inductive approach, let \(a^1_q=a_{a_q+2}\). While feasible, let \(a^1_{q+1}=a_{a_q+1}\), \(a^1_{q+2}=a_{a_q}\) and so on. In this case, let \(b^1_q=b_{b_q+2}\), \(b^1_{q+1}=b_{b_q+1}\), \(b^1_{q+2}=b_{b_q}\) and so on. Now, \(a^1_{a^1_q+2}\cdots a^1_q+\psi(b^1_{b^1_q+2}\cdots b^1_q)\) equals a constant string, where \(a^1_{a^1_q+2}\cdots a^1_q\) is an ascent and \(a^1_{a^1_q+2}=b^1_{b^1_q+2}\). The continuation of this procedure produces a subsequent string \(a^2_q\) and so on, until what remains to reach the leftmost entry of \(\alpha\) is smaller than the needed space for the procedure itself to continue, in which case, a remaining initial ascent is shared by both \(\alpha\) and \(\beta\). This allows to form the left-hand side of \(\alpha^p=\beta\) by concatenation.

Conflict of Interests

The author declares no conflict of interest.

1. Introduction

In this paper, we are concerned with simple finite graphs, without direction, multiple, or weighted edges, and without self-loops. Let \(G=(V,E)\) be such a graph and \(\lambda \in \mathbb{N}\). A mapping \(f : V (G)\longrightarrow \{1, 2,...,\lambda\}\) is called a \(\lambda\)-proper coloring of \(G\) if \(f(u) \neq f(v)\) whenever the vertices \(u\) and \(v\) are adjacent in \(G\). A color class of this coloring is a set consisting of all those vertices assigned the same color. If \(f\) is a proper coloring of \(G\) with the coloring classes \(V_1, V_2,..., V_{\lambda}\) such that every vertex in \(V_i\) has color \(i\), then we write simply \(f = (V_1,V_2,...,V_{\lambda})\). The chromatic number \(\chi(G)\) of \(G\) is the minimum number of colors needed in a proper coloring of a graph. The concept of a graph coloring and chromatic number is very well-studied in graph theory. A dominator coloring of \(G\) is a proper coloring of \(G\) such that every vertex of \(G\) dominates all vertices of at least one color class (possibly its own class), i.e., every vertex of \(G\) is adjacent to all vertices of at least one color class. The dominator chromatic number \(\chi_d(G)\) of \(G\) is the minimum number of color classes in a dominator coloring of \(G\).

The concept of dominator coloring was introduced and studied by Gera, Horton and Rasmussen [1]. Let \(G\) be a graph with no isolated vertex, then the total dominator coloring (TD-coloring) is a proper coloring of \(G\) in which each vertex of the graph is adjacent to every vertex of some (other) color class [2]. The total dominator chromatic number (TDC-number), \(\chi_d^t(G)\) of \(G\) is the minimum number of color classes in a TD-coloring of \(G\). The TDC-number of a graph is related to its total domination number. A total dominating set of \(G\) is a set \(S\subseteq V(G)\) such that every vertex in \(V(G)\) is adjacent to at least one vertex in \(S\). The total domination number of \(G\), denoted by \(\gamma_t(G)\), is the minimum cardinality of a total dominating set of \(G\). A total dominating set of \(G\) of cardinality \(\gamma_t(G)\) is called a \(\gamma_t(G)\)-set.

The literature on the subject on total domination in graphs was surveyed and detailed in the book [3]. It is not hard to see that for every graph \(G\) with no isolated vertex, \(\gamma_t(G) \leq \chi_d^t(G)\). Computation of the TDC-number is NP-complete. The TDC-number of some operations of two graphs was studied in [2]. Also Henning in [4] established the lower and upper bounds on the TDC-number of a graph in terms of its total domination number. He showed that, for every graph \(G\) with no isolated vertex satisfies \(\gamma_t(G) \leq \chi_d^t (G)\leq \gamma_t(G) + \chi(G)\). The properties of TD-colorings in trees was studied in [4]. Trees \(T\) with \(\gamma_t(T) =\chi_d^t(T)\) was characterized in [4].

The join \(G = G_1 + G_2\) of two graph \(G_1\) and \(G_2\) with disjoint vertex sets \(V_1\) and \(V_2\) and edge sets \(E_1\) and \(E_2\) is the graph union \(G_1\cup G_2\) together with all the edges joining \(V_1\) and \(V_2\). For two graphs \(G = (V,E)\) and \(H=(W,F)\), the corona \(G\circ H\) is the graph arising from the disjoint union of \(G\) with \(| V |\) copies of \(H\), by adding edges between the \(i\)th vertex of \(G\) and all vertices of \(i\)th copy of \(H\).

In this paper, we continue the study of TD-colorings in graphs. We compute the TDC-number of some specific graphs in the Section 2. In Section 3, we study TD-chromatic number of corona and join of graphs.

Total dominator chromatic number of specific graphs

In this section, we consider the specific graphs and compute their TDC-numbers. First we need the TDC-number of path and cycle graph. Note that the value of TDC-number of paths and cycles which have computed in [5] are lower and upper bounds for \(\chi_d^t(P_n)\), \(\chi_d^t(C_n)\) and are not the exact value. For example by formula in [5], \(\chi_ d^t (P_{60})=40\) which is not true and the correct value is \(32\) which can obtain by the following theorem.

Theorem 1. If \(P_n\) is the path graph of order \(n\geq 8\), then \[ \chi_d^t(P_n)=\left\{ \begin{array}{ll} {\displaystyle 2k+2}& \quad\mbox{if \(n=4k\), }\\ {\displaystyle 2k+3}& \quad\mbox{if \(n=4k+1\),}\\ {\displaystyle 2k+4}& \quad\mbox{if \(n=4k+2\), \(n=4k+3\).}\\ \end{array} \right. \] Also \( \chi _ d^t (P_3)=2\), \( \chi _ d^t (P_4)= 3\), \(\chi _ d^t (P_5)= \chi _ d^t (P_6)=4\) and \( \chi _ d^t (P_7)=5\).

Proof. It is easy to show that \( \chi _ d^t (P_3)=2\), \( \chi _ d^t (P_4)= \chi _ d^t (P_5)=3\), \( \chi _ d^t (P_6)=4\) and \( \chi _ d^t (P_7)=5\). Now let \(n\geq 8\). First we show that for each four consecutive vertices we have to use at least two new colors. Consider Figure 1. We have two cases. If we give an old color to \(v_{i+1}\), then we need to give a new color to \(v_{i+2}\) and \(v_{i+3}\) to have a TD-coloring. Also if we give a new color to \(v_{i+1}\), then we have to give a new color to \(v_{i+2}\) or \(v_{i}\) to have a TD-coloring. So we need at least two new colors in every four consecutive vertices.
Suppose that \(n=4k\), for some \(k\in \mathbb{N}\). We give a TD-coloring for the path \(P_{4k}\) which use only two new colors in every four consecutive vertices. Define a function \(f_0\) on the vertices of \(P_{4k}\), i.e., \(V(P_{4k})\) such that for any vertex \(v_i\), \[f_0(v_i)=\left\{ \begin{array}{ll} {\displaystyle 1} & \quad\mbox{if \(i=1+4k\),}\\ {\displaystyle 2} & \quad\mbox{if \(i=4k\),} \end{array} \right.\] and for any \(v_i\) , \(i \neq 4k\) and \(i\neq 4k+1\), \(f_0(v_i)\) is a new number. Then \(f_0\) is a TD-coloring of \(P_{4k}\) with the minimum number \(2k+2\).

If \(n=4k+1\), for some \(k\in \mathbb{N}\), then first color the \(4k-4\) vertices using \(f_0\). Now for the rest of vertices define \(f_1(v_{4k-3})=1\), \(f_1(v_{4k-2})=2k+1\), \(f_1(v_{4k-1})=2k+2\), \(f_1(v_{4k})=2k+3\) and \(f_1(v_{4k+1})=2\). Since for every five consecutive vertices we have to use at least three new colors, so \(f_1\) is a TD-coloring of \(P_{4k+1}\) with the minimum number \(2k+3\).
If \(n=4k+2\), for some \(k\in \mathbb{N}\), then first color the \(4k-4\) vertices using \(f_0\). Now for the rest of vertices define \(f_2(v_{4k-3})=1\), \(f_2(v_{4k-2})=2k+1\), \(f_2(v_{4k-1})=2k+2\), \(f_2(v_{4k})=2k+3\), \(f_2(v_{4k+1})=2k+4\) and \(f_2(v_{4k+2})=2\). Since for every six consecutive vertices we have to use at least four new colors, so \(f_2\) is a TD-coloring of \(P_{4k+2}\) with the minimum number \(2k+4\).
If \(n=4k+3\), for some \(k\in \mathbb{N}\), then first color the \(4k-4\) vertices using \(f_0\). Now for the rest of vertices define \(f_3(v_{4k-3})=1\), \(f_3(v_{4k-2})=2k+1\), \(f_3(v_{4k-1})=2k+2\), \(f_3(v_{4k})=2\), \(f_3(v_{4k+1})=2k+3\), \(f_3(v_{4k+2})=2k+4\) and \(f_3(v_{4k+2})=2\). Then \(f_3\) is a TD-coloring of \(P_{4k+2}\) with the minimum number \(2k+4\). Therefore we have the result.

Theorem 2. Let \(C_n\) be the cycle graph of order \(n\geq 8\). Then \[ \chi _ d^t (C_n)=\left\{ \begin{array}{ll} {\displaystyle 2k+2} & \quad\mbox{if \(n=4k\),}\\ {\displaystyle 2k+3}& \quad\mbox{if \(n=4k+1\),}\\ {\displaystyle 2k+4}& \quad\mbox{if \(n=4k+2\), \(n=4k+3\).} \end{array} \right. \] Also \( \chi _ d^t (C_3)=3\), \( \chi _ d^t (C_4)=2\), \( \chi _ d^t (C_5)= \chi _ d^t (P_6)=4\) and \( \chi _ d^t (C_7)=5\).

Proof. Observe that \(\chi_d^t(C_3)=3\), \( \chi_d^t (C_4)=2\), \( \chi_d^t (C_5)= \chi_d^t (C_6)=4\) and \( \chi_d^t (C_7)=5\). Now let \(n\geq 8\). It is sufficient to give a TD-coloring to the \(P_n\) as we see in the proof of the Theorem 1, then we add an edge between vertices \(v_1\) and \(v_n\). Then we have a TD-coloring for \(C_n\) and the result follows.

From Theorems 1 and 2, we have the following corollary.

Corollary 1. For every \(n\geq6\), \(\chi _ d^t (P_n)=\chi _ d^t (C_n)\).

Here we consider the ladder graph. We need the definition of Cartesian product of two graphs. Given any two graphs \(G\) and \(H\), we define the Cartesian product, denoted \(G\Box H\), to be the graph with vertex set \(V(G)\times V(H)\) and edges between two vertices \((u_1, v_1)\) and \((u_2,v_2)\) if and only if either \(u_1 = u_2\) and \(v_1v_2 \in E(H)\) or \(u_1u_2 \in E(G)\) and \(v_1 = v_2\).
Let \(n\geq 2\) be a natural number. The \(n\)-ladder graph can be defined as \(P_2\Box P_n\) and denoted by \(L_n\). Figure 2 shows a TD-coloring of ladder graphs.

Theorem 3. For every \(n\geq 2\), \[ \chi_d^t(L_n)=\left\{ \begin{array}{lr} {\displaystyle n+1}&\quad\mbox{if \(n\) is odd,}\\ {\displaystyle n}& \quad\mbox{if \(n\) is even.} \end{array} \right. \]

Proof. Let \(x_{ij}\) be a vertex of ladder graph in \(i\)-th row and \(j\)-th column (\(1\leq i\leq 2\) and \(1\leq j\leq n\)). If \(n=2k+1\) for some \(k\in \mathbb{N}\), then we color the vertex \(x_{1j}\) with color \(j\) and for vertices \(x_{2j}\) we assign the color \(2j+2\) for \(x_{2(2j+1)}\) and color \(2j-1\) for vertex \(x_{2(2j)}\) (Figure 2). This coloring gives a TD-coloring for \(L_n\) and this method warranty the least number of used colors. So \(\chi_d^t(L_{2k+1})=2k+2\). With similar argument we have the result for even \(n\).

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Click edit button to Here, we generalize the ladder graph \(P_2\Box P_n\) to grid graphs \(P_n\Box P_m\). The following theorem gives the TDC-number of grid graphs:

Theorem 4. Let \(m,n\geq 2\). The TDC-number of grid graphs \(P_n\Box P_m\) is, \[ \chi_d^t(P_n\Box P_m)=\left\{ \begin{array}{lr} {\displaystyle k \chi_d^t(P_n\Box P_2)=k\chi_d^t(L_n)}&\quad\mbox{if \(m=2k\) and \(n=2s\),}\\ {\displaystyle k\chi_d^t(L_n)+\chi_d^t(P_n)}& \quad\mbox{if \(m=2k+1\) and \(n=2s\),}\\ {\displaystyle s\chi_d^t(L_m)+\chi_d^t(P_m)}& \quad\mbox{if \(m=2k\) and \(n=2s+1\),}\\ {\displaystyle \chi_d^t(P_{n-1}\Box P_{m-1})+\chi_d^t(P_{m+n-1})}& \quad\mbox{if \(m=2k+1\) and \(n=2s+1\).} \end{array} \right. \]

Proof. We prove two first cases. The proof of another cases are similar. Suppose that \(m=2k\) and \(n=2s\), for some \(k\) and \(s\). We use induction on \(m\).
Case 1. If \(m=2\) and \(n=2s\), then we have a ladder and the result follows from Theorem 3. For \(m=2\), as you see in Figure 3, we have two \(L_n\) as subgraphs of \(4\times n\) grid graph. Since in TD-coloring of \(4\times n\) grid graph, we can not use the colors of vertices in the first ladder, for the second ladder, so we need \(2\chi_d^t(L_n)\) colors. It is easy to see that we cannot use less colors. Since in the \(P_n\Box P_{2k}\), there are exactly \(k\) ladder \(L_n\) as subgraphs, we have the result by induction hypothesis.this html

Case 2. Now suppose that \(n=2s\) and \(m=2k+1\). First for TD-coloring of \(P_n\Box P_{2k}\), by Case 1, we need \(k\chi_d^t(L_n)\) colors. It remains to color a path \(P_n\). Therefore we need \(k\chi_d^t(L_n)+\chi_d^t(P_n)\) colors to obtain a TD-coloring of \(P_n\Box P_m\). By the same argument in the proof of Theorem 3 we conclude that less colors cannot used for TD-coloring of this graph.

Now we consider graphs with specific construction. Let \(G\) be a connected graph constructed from pairwise disjoint connected graphs \(G_1,...,G_k\) as follows: Select a vertex of \(G_1\), a vertex of \(G_2\), and identify these two vertices. Then continue in this manner inductively. Note that the graph \(G\) constructed in this way has a tree-like structure, the \(G_i\)'s being its building stones (see Figure 4). Usually say that \(G\) is obtained by point-attaching from \(G_1,..., G_k\) and that \(G_i\)'s are the primary subgraphs of \(G\). A particular case of this construction is the decomposition of a connected graph into blocks (see [6]).

As an example consider the graph \(Q(m, n)\) constructed in the following manner: consider the graph \(K_m\) and \(m\) copies of \(K_n\) (see [6]). By definition, the graph \(Q(m, n)\) is obtained by identifying each vertex of \(K_m\) with a vertex of a unique \(K_n\), see Figure 4. The following theorem gives the TDC-number of \(Q(m,n\)):

Theorem 5. Let \(m,n\geq 2\) be integers. For the graph \(Q(m,n)\) we have: $$\chi_d^t(Q(m,n))=m+n-1.$$

Proof. We need colors \(1,2,\ldots,m\) to color the complete graph \(K_m\). Now for the rest of vertices of \(Q(m,n)\), we use colors \(m+1,m+2,\ldots,m+n-1\) in each \(K_n\). This gives a TD-coloring for \(Q(m,n)\). We shall show that we are not able to have TD-coloring with less colors. Suppose that we omit one color class between numbers \(m+1,m+2,\ldots,m+n-1\). Consider one copy of \(K_n\) and simply call it \(G_1\) and call the vertex which has no color as \(v\). We have to use one color from the numbers \(1,2,\ldots,m\) to color \(v\). Suppose that this color is \(i\), where \(1\leq i\leq m\). Then there is another copy of \(K_n\), say \(G_2\) which has a vertex with color \(i\) (point attached vertex). Also there is \(w \in V(G_2)\) such that has no color. We cannot color \(w\) with \(m+1,m+2,\ldots,m+n-1\). Since \(w\) is not adjacent to \(v\), so it cannot get an arbitrary color from \(1,2,\ldots,m\). Therefore we have to change the color of a vertex \(s\in V(G_2)-\{w\}\) to \(j \in \{1,2,\ldots,m\}\). But \(w\) is not adjacent to the vertex with color \(j\) in \(K_m\). So we cannot give the vertex \(w\) any color. By the same argument we cannot omit two color classes and so on. Therefore \(\chi_d^t(Q(m,n))=m+n-1\).

Let to consider a special cases of point attaching of \(k\) graphs. Let \(G_1,G_2,..., G_k\) be a finite sequence of pairwise disjoint connected graphs and let \(x_i, y_i\in V (G_i)\). The link \(G\) of the graphs \(\{G_i\}_{i=1}^k\) with respect to the vertices \(\{x_i, y_i\}_{i=1}^k\) is obtained by adding an edge which connect the vertex \(y_i\) of \(G_i\) with the vertex \(x_{i+1}\) of \(G_{i+1}\) for all \(i = 1, 2,..., k-1\), see Figure 5, [6].

Here we shall study the total dominator coloring of families of graphs which obtained by point attaching from \(G_1,...,G_k\).

Theorem 6. Let \(G\) be a graph obtained by point attaching from \(G_1,...,G_k\). Then $$\chi_d^t(G)\leq\chi_d^t(G_1)+\chi_d^t(G_2)+\ldots + \chi_d^t(G_n).$$

Proof. Since we can use numbers \(1,2,\ldots, \chi_d^t(G_1)\) for \(G_1\) and use numbers \( \chi_d^t(G_1)+1, \chi_d^t(G_1)+2,\ldots, \chi_d^t(G_1)+ \chi_d^t(G_2)\) for \(G_2\) and continue this process, then we have a TD-coloring . Therefore we have the result.

We shall show that the upper bound in Theorem 6 is sharp. For this reason, we consider a special kind of link graph has shown in Figure 5.

Theorem 7. For the graph \(L_{6,2,n}\) in Figure 5 we have: $$\chi_d^t(L_{6,2,n})=4n.$$

Proof. We color the vertices of \(L_{6,2,n}\) with numbers \(1,2,3,...,4n\), as shown in the Figure 5. Observe that, we need \(4n\) color for TD-coloring. We shall show that we are not able to have TD-coloring with less colors. Note that \(L_{6,2,1}\) is \(C_6\) and \(\chi_d^t(C_6)=4\). Now we consider \(L_{6,2,2}\). Two kinds of coloring of \(L_{6,2,2}\) has shown in Figure 6.

We show that \(\chi_d^t(L_{6,2,2})=8\). If we want to omit one color class and use another color, then we cannot have a TD-coloring. For example if we delete color \(5\), then

• (i) We cannot use color \(1\), since a vertex with color \(2\) exists which is not adjacent to the mentioned vertecis.
• (ii) We cannot use colors \(6,7\) (or color \(2\) in the right figure), since the coloring is proper.
• (iii) We cannot use color \(3\) or \(4\), because the vertices with these colors are not adjacent with the mentioned vertices.
• (iv) We cannot use color \(8\) because the vertex with color \(6\) is not adjacent to the vertex with color \(8\).

Argument about the rest of the colors is the same. So we have \(\chi_d^t(L_{6,2,2})=8\). Using this inductively method have the result for \(L_{6,2,n}\).

Here we investigate the relation of TDC-number of a graph \(G\) with TDC-number of \(G-e\), where \(e\in E(G)\):

Theorem 8. If \(G\) is a connected graph, and \(e=vw\in E(G)\) is not a bridge, then $$\chi_d^t(G-e)\leq \chi_d^t(G)+2 .$$

Proof. Suppose that the vertex \(v\) has color \(i\) and the vertex \(w\) has color \(j\). We have the following cases:
Case 1. The vertex \(v\) does not use the color class \(j\) and the vertex \(w\) does not use the color class \(i\) in the TD-coloring of \(G\). So the TD-coloring of \(G\) gives a TD-coloring of \(G-e\). Therefore \(\chi_d^t(G-e)\leq \chi_d^t(G)\). In this case \(\chi_d^t(G-e)=\chi_d^t(G)\).
Case 2. The vertex \(v\) uses the color class \(j\) but the vertex \(w\) does not use the color class \(i\) in the TD-coloring of \(G\). Since the vertex \(v\) used the color class \(j\) for The TD-coloring then we have two cases:

• (i) If the vertex \(v\) has some adjacent vertices which have color \(j\), then we give the new color \(l\) to all of these vertices and this coloring is a TD-coloring for \(G-e\).
• (ii) If any other vertex does not have color \(j\), since \(G-e\) is a connected graph, then there exists a vertex \(s\) which is adjacent to \(v\). Now we give the vertex \(s\) a new color \(l\) and this coloring is a TD-coloring for \(G-e\).

So for the Case 2, we have \(\chi_d^t(G-e)= \chi_d^t(G)+1\).
Case 3. The vertex \(v\) uses the color class \(j\) and the vertex \(w\) uses the color class \(i\) in the TD-coloring of \(G\). We have three cases:

• (i) There are some vertices which are adjacent to vertex \(v\) and have color \(j\). Then we color all of them with color \(l\), and there are some vertices which are adjacent to the vertex \(w\) and have color \(i\). We color all of them with color \(k\). So this is a TD-coloring for \(G-e\).
• (ii) Any other vertex does not have color \(j\). Then we do the same as Case 2 (ii) and there are some vertices which are adjacent to the vertex \(w\) and have color \(i\). Then we do the same as Case 3 (i).
• (iii) Any other vertex does not have colors \(i\) and \(j\). Then we do the same as Case 2 (ii) and use two new colors \(l\) and \(k\).

So we have \(\chi_d^t(G-e)\leq \chi_d^t(G)+2\).

3. TDC-number of corona and join of graphs

In this section, we study the TDC-number of corona and join of two graphs. In the following theorem, we consider graphs of the form \(G\circ H\) and study their TDC-numbers:

Theorem 9.

• (i) For every connected graph \(G\), \(\chi_d^t(G\circ K_1)=|V(G)|+1\),
• (ii) For every two connected graphs \(G\) and \(H\), $$\chi_d^t(G\circ H)\leq \chi_d^t(G)+|V(G)|\chi_d^t(H).$$
• (iii) For every two connected graphs \(G\) and \(H\), $$\chi_d^t(G\circ H)\leq |V(G)|+|V(H)|.$$

Proof.

• (i) We color all vertices of graph \(G\) with numbers \(\{1,2,...,|V(G)|\}\) and all pendant vertices with another color, say, \(|V(G)|+1\). It is easy to check that we are not able to have TD-color of \(G\circ K_1\) with less color. Therefore we have the result.
• (ii) For TD-coloring of \(G\) and \(H\), we need \(\chi_d^t(G)\) and \(\chi_d^t(H)\) colors, respectively. We observe that if we use \(\chi_d^t(G)+|V(G)|\chi_d^t(H)\) colors, then we have a TD-coloring of \(G\circ H\). So \(\chi_d^t(G\circ H)\leq \chi_d^t(G)+|V(G)|\chi_d^t(H)\).
• (iii) We color the vertices of \(G\), by \(|V(G)|\) colors and for every copy of \(H\), we use \(|V(H)|\) another colors. We observe that this coloring gives a TD-coloring of \(G\circ H\). So \(\chi_d^t(G\circ H)\leq |V(G)|+|V(H)|.\)

Remark 1 The upper bound for \(\chi_d^t(G\circ H)\) in Theorem 9(iii) is a sharp bound. As examples, for the graph \(C_4\circ K_2\) and \(K_2\circ K_3\) we have the equality (Figure 7).

Here, we state and prove a formula for the TDC-number of join of two graphs:

Theorem 10. Let \(G\) and \(H\) be two connected graphs, \(\vert V(G) \vert\geq 2 \) and \(\vert V(H) \vert\geq 2\) , then $$\chi_d^t(G+H)=\chi_d^t(G)+\chi_d^t(H).$$

Proof. For the TD-coloring of \(G+H\), the colors of vertices of \(G\) cannot be used for the coloring of vertices of \(H\), and the colors of the vertices of \(H\) cannot use for coloring of the vertices of \(G\), so $$\chi_d^t(G+H)\geq \chi_d^t(G)+\chi_d^t(H).$$ Now, it suffices to consider the coloring of \(G\) and the coloring of \(H\) in the TD-coloring of \(G+H\). Therefore, we have the result.

Acknowledgments

The authors would like to express their gratitude to the referees for their careful reading.

Author Contributions

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

Conflict of Interests

The authors declare no conflict of interest.