Introduction to total dominator edge chromatic number

TL;DR

This paper introduces the total dominator edge chromatic number, a new graph coloring parameter, and explores its properties, computations for specific graphs, and effects of graph modifications.

Contribution

It defines the TDEC-number, studies its properties, and analyzes its behavior under various graph operations and subdivisions, providing foundational insights.

Findings

Computed TDEC-number for specific graphs

Analyzed effects of graph modifications on TDEC-number

Studied TDEC-number for subdivided graphs

Abstract

We introduce the total dominator edge chromatic number of a graph $G$. A total dominator edge coloring (briefly TDE-coloring) of $G$ is a proper edge coloring of $G$ in which each edge of the graph is adjacent to every edge of some color class. The total dominator edge chromatic number (briefly TDEC-number) $\chi'^t_d(G)$ of $G$ is the minimum number of color classes in a TDE-coloring of $G$. We obtain some properties of $\chi'^t_d(G)$ and compute this parameter for specific graphs. We examine the effects on $\chi'^t_d(G)$ when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivison of $G$ and study TDEC-number of this kind of graphs.