Equitable Dominator Coloring of Graphs

TL;DR

This paper introduces equitable dominator coloring, a new graph coloring variant combining dominator and equitable coloring, and explores its properties and bounds for basic graph classes.

Contribution

It defines equitable dominator coloring and determines its chromatic number for basic graph classes, expanding the theory of graph colorings.

Findings

Equitable dominator chromatic number is established for certain graph classes.

Bounds for the equitable dominator chromatic number are discussed.

The concept extends existing graph coloring theories.

Abstract

This paper introduces a new variant of domination-related coloring of graphs, which is a combination of their dominator coloring and equitable coloring called the equitable dominator coloring. An equitable coloring is a proper coloring in which the number of vertices in each color class differs by at most one. In this newly introduced coloring, an additional condition of equitability is added to the existing concept of dominator coloring. The minimum number of colors used in this coloring is called the equitable dominator chromatic number, represented by $\chi_{ed}(G)$. The concept of equitable dominator coloring is explored for basic graph classes. The equitable dominator chromatic number is obtained for the same and some observations on the bounds are made in this paper.