On the edge r-dynamic chromatic number of some related graph operations

Abstract

All graphs in this paper are simple, nontrivial, connected and undirected. By an edge proper k-coloring of a graph G, we mean a map c : E(G) ! S, where jSj = k, such that any two adjacent edges receive different colors. An edge r-dynamic k-coloring is a proper k-coloring c of G such that jc(N(uv))j min (r; d(u) + d(v) 􀀀 2) for each edge uv in V (G), where N(uv) is the neighborhood of uv and c(S) = c(uv) : uv2S for an edge subset S. The edge r-dynamic chromatic number, written as r(G), is the minimum k such that G has an edge r-dynamic k-coloring. In this paper, we will determine the edge coloring r-dynamic number of a comb product of some graph, denote by G D H. Comb product of some graph is a graph formed by two graphs G and H, where each edge of graph G is replaced by which one edge of graph H.