Neighbor product distinguishing total colorings of 2-degenerate graphs

Abstract

A total-k-neighbor product distinguishing-coloring of a graph G is a mapping $$\phi : V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}$$ such that (1) any two adjacent or incident elements in $$V(G)\cup E(G)$$ receive different colors, and (2) for each edge $$uv\in E(G)$$, $$f_{\phi }(u)\ne f_{\phi }(v)$$, where $$f_{\phi }(x)$$ denotes the product of the colors assigned to a vertex x and its incident edges under $$\phi $$. The smallest integer k for which such a coloring of G exists is denoted by $$\chi ^{\prime \prime }_{\prod }(G)$$. In this paper, by using the famous Combinatorial Nullstellensatz, we show that if G is a 2-degenerate graph with maximum degree $$\varDelta (G)$$, then $$\chi ^{\prime \prime }_{\prod }(G) \le \max \{\varDelta (G)+2,7\}$$. Our results imply the results on $$K_4$$-minor free graphs with $$\varDelta (G)\ge 5$$ (Li et al. in J Comb Optim 33:237–253, 2017).