Total dominator coloring number of middle graphs

Abstract

A total dominator coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this paper, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitly the total dominator chromatic number of the middle graph of several known families of graphs.

Total domination number of middle graphs

“…Following our previous work [12][13][14][15], in this paper we study the total outer-connected domination number of the middle graph of a simple graph.…”

Total dominator total chromatic numbers of cycles and paths

“…Motivated by the relation between coloring and total dominating, the concept of total dominator coloring in graphs introduced in [8] by Kazemi, and extended in [5,7,9,10,13].…”

“…13 , e 23 , e 45 , e 9(10) , e (11)(12) }, V 2 = {v 5 , v 7 , v 9 , e 12 , e 34 , e(10)(11) , e (12)(13) }, V 3 = {v 4 , e 56 }, V 4 = {v 10 , e 89 }, V 5…”

Total domination number of middle graphs

“…The concept of total domination in graph theory was first introduced by Cockayne, Dawes and Hedetniemi in [3] and it has been studied extensively by many researchers in the last years, see for example [5], [6], [7], [13], [8], [10], [12] and [14]. The literature on this subject has been surveyed and detailed in the recent book [7].…”