The Dominator Edge Coloring of Graphs

Abstract

Let $G$ be a simple graph. A dominator edge coloring (DE-coloring) of $G$ is a proper edge coloring in which each edge of $G$ is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of $G$ is the minimum number of color classes among all dominator edge colorings of $G$ , denoted by ${\chi }_d^{\prime }\left(G\right)$ . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between $G$ and the operations of $G$ .