## INTRODUCTION TO TOTAL DOMINATOR EDGE CHROMATIC NUMBER

## INTRODUCTION TO TOTAL DOMINATOR EDGE CHROMATIC NUMBER

*S. Alikhani, N. Ghanbari*

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## Abstract

We introduce the total dominator edge chromatic number of a graph G. A total dominator edge coloring (brie y 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 (briey TDEC-number) 0t d (G) of G is the minimum number of color classes in a TDE-coloring of G. We obtain some properties of 0t d (G) and compute this parameter for speci c graphs. We examine the e ects on 0t d (G) when G is modi ed 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.

## Keywords

total dominator edge chromatic number; vertex removal; k-subdivision.