On Total r-Dynamic Coloring of Several Classes of Graphs and Their Related Operations

Abstract

All graphs in this paper are simple, connected and undirected. Let r, k be natural numbers. By a proper k-coloring of a graph G, we mean a map c : V (G) → S, where |S| = k, such that any two adjacent vertices receive different colors. A total r-dynamic coloring is a proper k-coloring c of G, such that ∀v ∈ V (G), |c(N(v))| ≥ min[r, d(v) + |N(v)|] and ∀uv ∈ E(G), |c(N(uv))| ≥ min[r, d(u) + d(v)]. The total r-dynamic chromatic number, written as χ ′′r (G), is the minimum k such that G has an r-dynamic k-coloring. Finding the total r-dynamic chromatic number is considered to be a NP-Hard problems for any graph. Thus, in this paper, we initiate to study χ′′ r (G) of several classes of graphs and and their related operations.