Double total dominator chromatic number of graphs

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph G with minimum degree at least 2 is a proper vertex coloring of G such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of G is called the double total dominator chromatic number, denoted by χdd(G). Therefore, we establish the close relationship between the double total dominator chromatic number χdd(G) and the double total domination number γ×2,t(G). We prove the NP-completeness of the problem. We also examine the effects on χdd(G) when G is modified by some operations. Finally, we discuss the χdd(G) number of square of trees by giving some bounds.