Adjacent Vertex Distinguishing Incidence Coloring of the Cartesian Product of Some Graphs

All graphs in this paper are simple, connected and undirected. We use V (G) and E(G) to denote the set of vertices and the set of edges of a graph G, respectively. And we denote the maximum degree of G by ∆(G). The undefined terminology can be found in [1]. Let G be a graph of order n. For any vertex u ∈ V (G), N(u) denotes the set of all vertices adjacent to vertex u. Obviously, d(u) is equal to |N(u)|. Let I(G) = {(v, e) ∈ V (G)× E(G) | v is incident with e} be the set of incidences of G. Two incidences (v, e) and (w, f) are said to be adjacent if one of the following holds: (1) v = w; (2) e = f ; (3) the edge vw equals e or f . Iv = {(v, vu) | u ∈ N(v)} and Av = {(u, uv) | u ∈ N(v)} are called the close-incidence set and far-incidence set of v, respectively.