On Equitable Coloring of Book Graph Families

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by atmost one. The notion of equitable coloring was introduced by Meyer in 1973. A proper $h-$colorable graph $K$ is said to be equitably h-colorable if the vertex sets of $K$ can be partioned into $h$ independent color classes $V_1, V_2,...,V_h$ such that the condition $\left|\left|V_i\right|-\left|V_j\right|\right| \leq 1$ holds for all different pairs of $i$ and $j$ and the least integer $h$ is known as equitable chromatic number of $K$. In this paper, we find the equitable coloring of book graph, middle, line and central graphs of book graph.


Introduction
The idea of equitable coloring was discovered by Meyer [4] in 1973. Hajmal and Szemeredi [3] proved that graph K with degree is equitable h-colorable, if h + 1. Later Equitable Coloring Conjecture for bipartite graphs was proved. Equitable vertex coloring of corona graphs is NP-hard.
The graphs considered here are simple. Vertex coloring is a particular case of Graph coloring. The collection of vertices receiving same color is known as color class. A proper h colorable graph K is said to be equitably h colorable if the vertex sets of K can be partitioned into h independent color classes V 1 ; V 2 ; :::; V h such that the condition jjV i j jV j jj 1 holds for all di¤erent pairs of i and j [1]. And the least integer h is known as equitable chromatic number of K [1]. Here we found equitable coloring of book graph, middle, line and central graphs of book graph.

Preliminaries
Line graph [2] of K, L(K) is attained by considering the edges of K as the vertices of L(K). The adjacency of any two vertices of L(K) is a consequence of the corresponding adjacency of edges in K.
Middle graph [5] of K, M (K) is attained by adding new vertex to all the edges of K. The adjacency of any two new vertices of M (K) is a consequence of the corresponding adjacency of edges in K or adjacency of a vertex and an edge incident with it.
Central graph [6] of K, C(K) is attained by the insertion of new vertex to all the edges of K and connecting any two new vertices of K which were previously non-adjacent.
The q-book graph is de…ned as the graph Cartesian product S (q+1) P 2 , where S q is a star graph and P 2 is the path graph.  where z, k s , l s and m s are the subdivision of the edges gh, gg s , hh s and g s h s respectively. Let us consider V [C(B q )] and the color set C = fc 1 ; c 2 ; :::; c q+1 g. Assign the equitable coloring by Algorithm B. Therefore,

On Equitable Coloring of Line Graph of Book Graph.
Order Order of B q is 2(q + 1) Number of incidents of B q is 3q + 1 Maximum degree of B q is q + 1 Minimum degree of B q is 2 Algorithm D Input: The value 'q'of B q , for q 3 Outcome: Equitably colored V (B q ) Procedure: start { for s = 1 to q { V a = fg s ; hg ; C(h)=1; C(g s ) = 1; } for s = 1 to q f V b = fg; h s g ; C(g) = 2; C(h s ) = 2; Let us consider the V (B q ) and the color set C = fc 1 ; c 2 g. Assign the equitable coloring by Algorithm D. Therefore, = (B q ) 2: And since, there exists a maximal induced complete subgraph of order 2 in B q (say path P 2 ). Therefore, = (B q ) 2 c 1 ; c 2 are independent sets of B q . And jjc i j jc j jj 1, for every di¤erent pair of i and j. Hence, = (B q ) = 2: