Total dominator chromatic number of k-subdivision of graphs

Abstract

Let G be a simple graph. A total dominator coloring of G, is a proper coloring of the vertices of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number χdt(G) of G, is the minimum number of colors among all total dominator coloring of G. For any k ∈ ℕ, the k-subdivision of G is a simple graph G1/k which is constructed by replacing each edge of G with a path of length k. In this paper, we study the total dominator chromatic number of k-subdivision of G.