Introduction to total dominator edge chromatic number

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) χʹtd (G) of G is the minimum number of color classes in a TDE-coloring of G. We obtain some properties of χʹtd (G) and compute this parameter for specific graphs. We examine the effects on χʹtd (G) when G is modified by operations on vertices and edges of G. Finally, we consider the k-subdivison of G and study TDEC-number of this kind of graphs.