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).