A study on near proper coloring of graphs

Abstract

An equitable coloring of a graph G is a proper vertex coloring in which the number of vertices in any two color classes are equal or almost equal. In this graph coloring, there is a partition of tasks into subsets which perform at the same time. Equitable coloring play an important role when there is a requirement of dividing a system with binary conand#64258;ict free subsystems with equal or newlinenearly equal elements. The non-availability of sufcient number of colors leads to diand#64256;erent defective coloring problems. An equitable near proper coloring of a graph G is an improper coloring in which the vertex set can be partitioned into k color classes V1, V2, . . . , Vk ; (1 lt k lt and#967;e(G)) such that the number of vertices in any two color classes diand#64256;er by at most one and the resulting monochromatic edges are minimised by restricting the number of color classes that can have adjacency among their own elements. The minimum number of monochromatic newlineedges obtained from an equitable near proper coloring of G is called equitable defective number. This study introduces the notion of equitable near proper coloring of a graph G and investigates the equitable defective number for a few graph classes. In this study, the equitable near proper coloring of various graph classes and derived graphs are discussed, and the corresponding equitable newlinedefective number for any k where 2 and#8804; k and#8804; and#967;e(G) and#8722; 1 is obtained.