Neighbor Product Distinguishing Total Coloring of Planar Graphs without 5-cycles

Given a simple graph G and a proper total-k-coloring φ from V (G) ∪ E(G) to {1, 2,…,k}. Let f(v) = φ(v)Π_uv∈E(G)φ(uv). The coloring φ is neighbor product distinguishing if f(u) ≠ f(v) for each edge uv ∈ E(G). The neighbor product distinguishing total chromatic number of G, denoted by \(\chi_{\Pi}^{\prime\prime}(G)\) $${\chi }_{\mathrm{\Pi }}^{\mathrm{\prime }\mathrm{\prime }}(G)$$ , is the smallest integer k such that G admits a k-neighbor product distinguishing total coloring. Li et al. conjectured that \(\chi_{\Pi}^{\prime\prime}(G)\leq \Delta(G)+3\) $${\chi }_{\mathrm{\Pi }}^{\mathrm{\prime }\mathrm{\prime }}(G)\le \mathrm{\Delta }(G)+3$$ for any graph with at least two vertices. Dong et al. showed that conjecture holds for planar graphs with maximum degree at least 10. By using the famous Combinatorial Nullstellensatz, we prove that if G is a planar graph without 5-cycles, then \(\chi_{\Pi}^{\prime\prime}(G)\leq \max\{\Delta(G)+2,12\}\) $${\chi }_{\mathrm{\Pi }}^{\mathrm{\prime }\mathrm{\prime }}(G)\le max\{\mathrm{\Delta }(G)+2,12\}$$ .