Neighbor product distinguishing total colorings

A total-[k]-coloring of a graph G is a mapping $$\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}$$ ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } such that any two adjacent elements in $$V (G) \cup E(G)$$ V ( G ) ∪ 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 $$f(u)\ne f(v)$$ f ( u ) ≠ f ( v ) , where $$uv\in E(G)$$ u v ∈ E ( G ) . By $$\chi ^{\prime \prime }_{\prod }(G)$$ χ ∏ ″ ( G ) , we denote the smallest value k in such a coloring of G. We conjecture that $$\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3$$ χ ∏ ″ ( 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 $$K_4$$ K 4 -minor free graph with $$\Delta (G)\ge 4$$ Δ ( G ) ≥ 4 , then $$\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2$$ χ ∏ ″ ( G ) ≤ Δ ( G ) + 2 .