Adjacent vertex distinguishing edge-colorings and total-colorings of the lexicographic product of graphs

Let G be a simple graph with vertex set V(G) and edge set E(G). An edge-coloring f of G is called an adjacent vertex distinguishing edge-coloring of G if Cf(u)≠Cf(v) for any uv∈E(G), where Cf(u) denotes the set of colors of edges incident with u. A total-coloring g of G is called an adjacent vertex distinguishing total-coloring of G if Sg(u)≠Sg(v) for any uv∈E(G), where Sg(u) denotes the set of colors of edges incident with u together with the color assigned to u. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of G is denoted by χa′(G) (resp. χat(G)). The lexicographic product of simple graphs G and H is simple graph G[H] with vertex set V(G)×V(H), in which (u,v) is adjacent to (u′,v′) if and only if either uu′∈E(G) or u=u′ and vv′∈E(H). In this paper, we consider these parameters for the lexicographic product G[H] of two graphs G and H. We give the exact values of χa′(G[H]) if (1) G is a complete graph of order n≥3 and H is a graph of order 2m≥4 with χa′(H)=Δ(H); (2) G is a tree of order n≥3 and H is a graph of order m≥3 with χa′(H)=Δ(H). We also obtain the exact values of χat(G[H]) if (1) G is a complete graph of order n≥2 and H is an empty graph of order m≥2, where nm is even; (2) G is a complete graph of order n≥2 and H is a bipartite graph of order m≥4 with bipartition (X,Y), where |X| and |Y| are even; (3) G is a cycle of order n≥3 and H is an empty graph of order m≥2, where nm is even; (4) G is a tree of order n≥3 and H is an empty graph of order m≥2.