On Equitable Coloring of Book Graph Families

M Barani,Venkatachalam M,K Rajalakshmi

doi:10.31801/cfsuasmas.769094

Abstract

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.

Highlights

The idea of equitable coloring was discovered by Meyer [4] in 1973
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 V1; V2; :::; Vh such that the condition jjVij jVjjj 1 holds for all di¤erent pairs of i and j [1]

Summary

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. Line graph [2] of K, L(K) is attained by considering the edges of K as the vertices of L(K).

Results

Conclusion