Neighbor product distinguishing total colorings

Abstract

A total-[k]-coloring of a graph G is a mapping $\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}$ such that any two adjacent elements in $V (G) \cup E(G)$ 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 $f(u)\ne f(v)$, where $uv\in E(G)$. By $\chi ^{\prime \prime }_{\prod }(G)$, we denote the smallest value k in such a coloring of G. We conjecture that $\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3$ for any simple graph with maximum degree $\Delta (G)$. 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 $K_4$-minor free graph with $\Delta (G)\ge 4$, then $\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2$.