Abstract

An equitable vertex coloring for the splitting of block circulant graphs is investigated. The block circulant graphs comprises block circulant matrices, where each block is itself a matrix. These blocks in each row are cyclically shifted one place to the right from those of the previous row. We approached such block circulant graphs in matrix representation and derived their independent sets using the neighbourhoods of each vertex. This classification makes the vertex coloring process to be simpler and equitable in most cases. In this framework, the equitable chromatic numbers are obtained for splitting on block circulant graphs, namely prism, antiprism, crossed and closed sun graphs.

1. Introduction

In this paper, we consider on block circulant graph. Circulant graphs remains an active area of research for decades because of their wider applications in computer science, coding theory and discrete mathematics. A circulant graph is a graph having a circulant matrix. It is a matrix in which the elements of each row are taken from those of the previous row moved one place to the right.

The few more practical applications of circulant networks are, it forms a basis of the structure in a model of small-world networks, an important class of interconnection networks in parallel and distributed computing, a base structure for discrete cellular neural networks and in coding theory for the construction of perfect error-correcting codes, etc. In 2012 Monakhova [1] studied the structural and communicative properties of circulant networks.

Heuberger [2] in 2003 completely characterized the planarity and calculated the chromatic number for circulant graphs. Discepoli et al. [3] in 2004 constructed two different heuristics to choose the best set of eigenvectors through which the correct coloration of the graph is found. Ilic and Basic [4] in 2010 provided upper and lower bounds on the chromatic number of integral circulant graphs with arbitrary number of divisors. Codenotti et al. [5] have shown that the computation of chromatic number for circulant graphs whose adjacency matrices are circulant, is an NP-hard problem. In 2009 Barajas and Serra [6] studied the regular chromatic number of circulant graphs. In 2017 Nicoloso and Pietropaoli [7] discussed an exact vertex colouring algorithm for 3-chromatic circulant graphs.

In 2013 Hongbo and Xiaofeng [8] calculated the characteristic polynomials and spectra of several special block circulant graphs. Garcia et al. [9] in 2019 provides a natural bijection between the associated graphs of generalized crowns and a particular family of block circulant graphs. Recently in 2022 Goedgebeur and Van Overberghe [10] presented algorithms to establish the new lower bounds and exact values on Ramsey numbers involving circulant and block-circulant graphs.

However, only few research works has been carried out in the chromatic number problem for circulant and block circulant graphs. It still remains an active area of research. This led to the idea of applying splitting on block circulant graphs and explore its equitable vertex coloring.

Sampathkumar et al. [11] introduced the concept of splitting graph of a graph denoted by . It is framed by adding to each vertex , a new vertex such that is adjacent to every vertex that is adjacent to in . In this work, the splitting is considered for family of prism graphs and closed sun graph.

Besides this circulant and block circulant graphs, the idea of bounds remains open for various other graph such as Y-index of graphs and Zagreb indices of graphs. In 2022, Maji et al. explored the test of having a high correlation with the physiochemical properties passed through the new chemical descriptors Y-index of graphs [12]. The new topological indices, such as the Zagreb index of graphs [13], have a good correlation with various physical attributes, chemical reactives, or biological properties that can be explored by equitable chromatic bounds.

1.1. Organization of the paper

This paper is organized as follows: The next section contains some basic definitions and important results of equitable coloring related to this paper. In section 3, the adjacency matrix of block circulant graphs, like closed sun, prism, Antiprism and crossed prism are structured. In section 4, the theorems with proofs is constructed for splitting of block circulant graphs discussed in previous section, and their equitable chromatic number are determined. Finally, the conclusion of this study with some future work related to this field is suggested in section 5.

2. Preliminaries

Graph coloring is one of the interesting concept and has wider applications in assignment, scheduling and balancing problems, etc. For example, consider the scheduling problem like collection of tasks to be done, which are represented as the vertices of the graph and the edges are connected between two conflicting tasks that should not be performed at the same time. When graph coloring is applied to such scheduling problem, the coloring on the vertices of this graph partitions the tasks into subsets that may be done at the same time. The balancing of such tasks to be performed properly in more or less equally within the available time slots is achieved by equitable coloring. Here we present some basic definitions related to this work.

A circulant matrix [14] is generated from the -vector by cyclically permuting its entries, and is of the form.

Definition 2.1. (see [14]). A circulant matrix is defined completely by the generating elements in its first row, which are cyclically shifted to the right by one position per row to form the subsequent rows. The circulant matrix is the set of all such matrices of order , which is denoted by .

Definition 2.2. (see [14]). A block circulant matrix of dimension is a matrix generated by the matrices each of dimension and is symbolized as . The block rows of are obtained by cyclically shifting the ’s as follows:The matrix array defined above is said to be a block circulant of type. The set of all such matrices is denoted by . Any matrix in a block circulant matrix need not be necessarily a circulant.

Definition 2.3. (see [11]). For each point of a graph , take a new point . Join to all points of adjacent to . The graph thus obtained is called the Splitting graph of .

Definition 2.4. [15]. The set of vertices of a graph can be colored with colors such that no edge joins the vertices of the same color and the cardinalities of the color sets differ by maximum value of one, then is said to be equitably -colorable.
The smallest integer for which is equitably -colorable is known as the equitable chromatic number and is denoted by .

Conjecture 2.5 (see [15]). For any connected graph , other than complete graph or odd cycle, then .
Bodleander and Fomin [16] proved both the problems of equitable -coloring and -bounded -coloring can be solved to optimality in polynomial time for graphs of bounded tree-width. In 2006, Furmańcyzk [17] studied the equitable coloring of Cartesian and tensor product of graphs. In 2013, Kaliraj et al. [18] investigated the equitable coloring’s on Myscielkian construction of wheels and bi-graphs. In 2021, Veninstine and Xavier [19] applied spectral cluster techniques to achieve equitable coloring for distributed web networks.
This paper focus on the study of block circulant graphs, splitting block circulant graphs and determining their equitable chromatic number. The embodying of splitting of graph with block circulant graph and obtaining its equitable chromatic number is a new approach. Throughout this paper, we propose the idea of identifying the neighbours of each -vertex in splitting block circulant graph by considering its adjacency matrix. The -independent sets of cardinal number 4 are formed by taking the difference between the neighbourhood sets.
An independent set [20] in a graph is a set of pairwise nonadjacent vertices. In this work the independent sets are represented as . The coloring is applied to these independent sets and verified for equitability. If it is not equitable, further some the independent sets are parted into subsets to make the coloring as equitable. In this work, we obtain the equitable chromatic number for spitting on graphs such as prism, antiprism, crossed prism, and closed sun graph.

3. Graphs with block circulant adjacency matrix

In this section, the matrix representation of considered block circulant graphs is constructed with examples. A graph with vertices having a central complete graph and an outer ring of vertices, each of which is joined to both end points in the closest outer edge of the central core is called a sun graph, denoted as .

The closed sun graph is obtained from sun graph by adding edges to the outer ring of vertices and is denoted as . this graph can be viewed as a block circulant graph by considering its adjacency matrix.

For example, consider a closed sun graph (Figure 1) and its adjacency matrix as follows.

The adjacency matrix can be represented in block matrix aswhere

Thus, is a block circulant of type . Hence the closed sun graph is a block circulant graph but not a circulant graph.

The prism graph [21] is a graph obtained from two disjoint cycles by connecting the vertices in the inner cycle with the corresponding vertices in the outer cycle. It is represented in this work as . Prism graph is also called as a circular ladder graph .

For example, consider a prism graph (Figure 2) and its adjacency matrix as follows.

For prism graph its adjacency matrix can be given in block matrix as

where

Here , and are circulant matrices. But the entire prism graph, is a block circulant graph of type (2, 2). Hence the prism graph is a block circulant graph and on the whole it is not a circulant graph.

The antiprism graph [22] is a graph having vertices on two cycles labelled with in the inner cycle, in the outer cycle. Join these two cycles with edges of the form , and such that and and (mod n) are adjacent. It is symbolized as .

For example, consider a antiprism graph as shown in Figure 3 and its adjacency matrix as follows.

The adjacency matrix can be represented in block matrix as

It is observed that and are separately circulant matrices. The full prism graph, is a block circulant graph with block circulant matrix of type . Hence the prism graph is a block circulant graph and is not a circulant graph.

A crossed prism graph [22] for positive even , is a graph obtained from two cycles and joining these cycles with edges and for . The crossed prism graph is denoted by .

For example, consider a crossed prism graph (Figure 4) and its adjacency matrix as follows.

The adjacency matrix of crossed prism graph can be written in block matrix aswhere

Thus, is a block circulant of type . Hence the crossed prism graph is a block circulant graph and not a circulant graph.

4. Equitable coloring in splitting of block circulant graphs

In this section, the splitting of graph is considered for block circulant graphs such as , , and . Their generalized adjacency matrix with blocks are constructed. The partition of independent sets is done through the difference of neighbourhood sets identified from the rows of the adjacency matrix. The equitable vertex coloring is established by applying minimum colors to the vertices of the independent sets. The figures depict the colors assigned to the vertices of each independent set in an equitable manner.

Theorem 4.1. The equitable vertex coloring for splitting of Prism graph is

Proof. Consider a splitting Prism graph (see Figure 5), which generates a graph with 24 vertices and 54 edges.
Generally, the splitting of block circulant Prism graph generates a graph with vertices and edges which is not a block circulant. Its generalized adjacency matrix can be structured in blocks as follows.
Consider the first row of the adjacency matrix corresponding to the vertex and columns to obtain the neighbourhood .The neighbourhood of the vertex are identified from columns having the entries as 1.The second row of the adjacency matrix corresponds to the neighbourhood of the vertexTherefore, Similarly the neighbourhood of the remaining vertices are obtained from the corresponding rows of the adjacency matrix of graph .The repetition of vertices in are refined by finding the difference of the neighbourhood sets which produces the independent sets .The coloring of the vertices is made through these independent sets.

Case. (i)If is odd,
Consider the coloring as a mapping function from , where .
When To achieve equitable coloring the set is divided into two subsets and such that and which implies . The coloring of the elements (vertices) in this two subsets are done as follows.Thus the colors assigned on the vertices of the graph through this method of mapping the function yields the two independent color classes and having colors 1 and 2 respectively. For instance see Figure 6.It is evident that , therefore satisfies the equitable condition. In this case .

Case (ii). If is even and or or
Consider the coloring as a mapping function from , where .
When It is clear that the difference between two color classes and is less than or equal to one, where and . As a consequence . Hence the equitable condition is satisfied by . Thus (see Figure 7).

Case (iii). If is odd,
The independent sets , and are split up into subsets for making the proper equitable vertex coloring. The set is separated into subsets , and , where , and . It implies that . The set is divided into subsets and , which implies , and hence . Likewise the set is partitioned into two subsets and . After splitting form the union of the sets , whereNext the coloring is mapped as a function fromWhen After the coloring the color sets are classified asThe bounds for this color partitions is and . It is observed that the difference between any two color classes vary by the extent value of one. Therefore it satisfies the equitable condition . Hence it is equitably 3-colorable (see Figure 8).

Case (iv). If is odd,
In this case the independent set is divided into , such that and . Also . Let . The set is partitioned into , such that , and hence which is represented as . Form the union of the sets , as
Consider the coloring as a mapping function from , where .
For Using this method the color sets are sorted asThe absolute value of grouping the color classes is and . Clearly . It is true that equitable condition holds for this case of graph. Hence (see Figure 9). □

Theorem 4.2. The equitable vertex coloring for splitting of Antiprism graph is

Proof. Consider a splitting Antiprism graph (see Figure 10), which generates a graph with 16 vertices and 48 edges.
Generally, the splitting of Antiprism graph emerges as a graph with vertices and edges which is not block circulant. Its adjacency matrix can be represented in matrix blocks as follows.
Each row in the adjacency matrix of this graph portray the adjacent vertices of each vertex resulting in their neighbourhood . The first row corresponds to the vertex shows the neighbouring elements which are represented with 1 in column-wise. The second row of the adjacency matrix unfolds the neighbouring elements of the vertex in this graph.Similarly the successive rows upto vertex of the adjacency matrix gives the remaining neighbourhood sets of the splitting on Antiprism graph.Further the independent subsets of the neighbourhood sets are obtained by considering the difference between the following neighbourhood sets.The vertices of the independent sets are colored through the mapping function , . The color sets are and .

Case 5. If is odd or even,
Consider the coloring as a mapping function from .
When Hence the vertices of the independent sets of the whole graph under this case are colored with only 3 colors (see Figure 11) and can be grouped into color classes and , where , and . The cardinal of the color sets are which implies the equitable condition holds for , . Hence for the splitting antiprism graphs in this type .

Case 6. (i) If is odd and
The independent sets , and are split into subsets for equitable vertex coloring. The set is separated into subsets and , where and . It implies that . The set is divided into subsets and , which implies , and hence . Also the set is partitioned into two subsets and such that . For the coloring to be equitable form the union of the sets , whereAfter these reformation the coloring is allocated as a function fromWhen It requires 4 colors (see Figure 12) and the color classesareThe number of vertices in the color sets are . Further the difference between any of these two color classes differs by the value of one. It confirms the equitable condition and is equitably 4-colorable.
(ii) If is odd and .
In this case the independent sets , and are divided into subsets. The set is partitioned into subsets and , where . The set is split into subsets and , hence . Also the set is parted into two subsets and such that . The coloring process is allocated after the union of the sets , whereThe coloring is done as a function ofFor In this way the vertices of the independent sets are allotted with 4 colors in an uniform manner to reach the equitable condition (see Figure 13). The color sets areThe cardinal number of the color sets are which proves that is equitable. In this case
(iii) When is even and a multiple of 4, also or
The coloring in this case is considered as a function such that .
For In this method all the vertices of the independent sets of this graph is colored equally by 4 colors (see Figure 14) with color classes partitions asand their cardinal numbers are . Thereof satisfies the equitable condition . Hence under both these cases or , .
(iv) When is even and not a multiple of 4, along with or
In this case the independent sets and are divided into subsets. The set is parted into subsets and , where . Also the set is partitioned into two subsets and such that .
The coloring is defined as a function .When It is observed that all the vertices under this case of the graph are colored equally with 4 colors (see Figure 15), thus grouping the color classes aswith cardinal numbers as . Hence it is suffice to prove the equitable condition . Also in these cases or , .

Theorem 4.3. The vertex coloring on splitting of crossed prism graph is equitably 2- colorable. .

Proof. Consider a splitting crossed prism graph (see Figure 16), which generates a graph with 24 vertices and 54 edges.
Generally, the crossed prism graph contains vertices and edges and the splitting of crossed prism graph promotes a graph with vertices and edges which is not block circulant on the whole matrix. The adjacency matrix of can be generalized in blocks as follows
The neighbouring elements of the vertices to this splitting graph corresponds to each row in the adjacency matrix. The first row relate to the vertex shows the neighbouring elements which are represented as 1 in column-wise. The second row present the neighbouring elements of the vertex in this graph.Consider the consecutive rows in the adjacency matrix of crossed prism graph up to vertex.The independent subsets of this graph is obtained by taking the difference of the neighbourhood sets as follows.The coloring of this graph is accomplished by allocating colors to the vertices of the independent sets. Define a function such that , where .Through this method it needs only 2 colors for coloring the splitting crossed prism graph and found to be equitably allotted (see Figure 17). Let the two color classes be and . Here , thus satisfies the equitable condition . Hence .

Theorem 4.4. The vertex coloring on splitting of closed sun graph is equitable. .

Proof. Consider a splitting closed sun graph (see Figure 18), which generates a graph with 16 vertices and 54 edges.
Generally, the block circulant closed sun graph contains vertices and edges and the splitting of closed sun graph has vertices and edges which is not block circulant on the whole matrix. The adjacency matrix of can be generalized in blocks as follows.
The neighbourhood for each vertices of the graph can be identified from the rows of the adjacency matrix of this graph. The first row corresponds to the vertex whose neighbouring elements are represented with 1 in column-wise. The neighbouring elements of the vertex are identified from the second row of the adjacency matrix.