Neighbor product distinguishing total colorings

Abstract

A total-[k]-coloring of a graph G is a mapping $$\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}$$ź:V(G)źE(G)ź{1,2,ź,k} such that any two adjacent elements in $$V (G) \cup E(G)$$V(G)źE(G) receive different colors. Let f(v) denote the product of the color of a vertex v and the colors of all edges incident to v. A total-[k]-neighbor product distinguishing-coloring of G is a total-[k]-coloring of G such that $$f(u)\ne f(v)$$f(u)źf(v), where $$uv\in E(G)$$uvźE(G). By $$\chi ^{\prime \prime }_{\prod }(G)$$źźź(G), we denote the smallest value k in such a coloring of G. We conjecture that $$\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3$$źźź(G)≤Δ(G)+3 for any simple graph with maximum degree $$\Delta (G)$$Δ(G). In this paper, we prove that the conjecture holds for complete graphs, cycles, trees, bipartite graphs and subcubic graphs. Furthermore, we show that if G is a $$K_4$$K4-minor free graph with $$\Delta (G)\ge 4$$Δ(G)ź4, then $$\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2$$źźź(G)≤Δ(G)+2.