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 χ_d^t(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 G^1/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.