Neighbor sum distinguishing total coloring of planar graphs without 4-cycles

Abstract

Let $G=(V,E)$ be a graph and $\phi : V\cup E\rightarrow \{1,2,\ldots ,k\}$ be a proper total coloring of G. Let f(v) denote the sum of the color on a vertex v and the colors on all the edges incident with v. The coloring $\phi$ is neighbor sum distinguishing if $f(u)\ne f(v)$ for each edge $uv\in E(G)$. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number of G, denoted by $\chi _{\Sigma }''(G)$. Pilśniak and Woźniak conjectured that $\chi _{\Sigma }''(G)\le \Delta (G)+3$ for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that $\chi _{\Sigma }''(G)\le \max \{\Delta (G)+2, 10\}$ for planar graph G without 4-cycles. The bound $\Delta (G)+2$ is sharp if $\Delta (G)\ge 8$.