Neighbor product distinguishing total colorings of 2-degenerate graphs

A total- k -neighbor product distinguishing-coloring of a graph G is a mapping $$\phi : V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}$$ ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } such that (1) any two adjacent or incident elements in $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) receive different colors, and (2) for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$f_{\phi }(u)\ne f_{\phi }(v)$$ f ϕ ( u ) ≠ f ϕ ( v ) , where $$f_{\phi }(x)$$ f ϕ ( x ) denotes the product of the colors assigned to a vertex x and its incident edges under $$\phi $$ ϕ . The smallest integer k for which such a coloring of G exists is denoted by $$\chi ^{\prime \prime }_{\prod }(G)$$ χ ∏ ″ ( G ) . In this paper, by using the famous Combinatorial Nullstellensatz, we show that if G is a 2-degenerate graph with maximum degree $$\varDelta (G)$$ Δ ( G ) , then $$\chi ^{\prime \prime }_{\prod }(G) \le \max \{\varDelta (G)+2,7\}$$ χ ∏ ″ ( G ) ≤ max { Δ ( G ) + 2 , 7 } . Our results imply the results on $$K_4$$ K 4 -minor free graphs with $$\varDelta (G)\ge 5$$ Δ ( G ) ≥ 5 (Li et al. in J Comb Optim 33:237–253, 2017).