On Domination Coloring in Graphs

TL;DR

This paper introduces the concept of domination coloring in graphs, studies its computational complexity, and explores its relationships with other graph invariants across various graph classes.

Contribution

It proves NP-Completeness of the domination coloring problem and provides bounds, characterizations, and relations with other graph parameters.

Findings

NP-Completeness of domination coloring problem.

Bounds and characterizations for specific graph classes.

Relations between domination chromatic number and other graph invariants.

Abstract

A domination coloring of a graph $G$ is a proper vertex coloring of $G$ such that each vertex of $G$ dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all domination colorings is called the domination chromatic number, denoted by $\chi_{dd}(G)$. In this paper, we study the complexity of this problem by proving its NP-Completeness for arbitrary graphs, and give general bounds and characterizations on several classes of graphs. We also show the relation between dominator chromatic number $\chi_{d}(G)$, dominated chromatic number $\chi_{dom}(G)$, chromatic number $\chi(G)$, and domination number $\gamma(G)$. We present several results on graphs with $\chi_{dd}(G)=\chi(G)$.