Neighbor sum distinguishing total colorings of planar graphs

Abstract

A total [k]-coloring of a graph $G$ is a mapping $\phi : V (G) \cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$ such that any two adjacent or incident elements in $V (G) \cup E(G)$ receive different colors. Let $f(v)$ denote the sum of the color of a vertex $v$ and the colors of all incident edges of $v$. A total $[k]$-neighbor sum distinguishing-coloring of $G$ is a total $[k]$-coloring of $G$ such that for each edge $uv\in E(G)$, $f(u)\ne f(v)$. By $\chi ^{''}_{nsd}(G)$, we denote the smallest value $k$ in such a coloring of $G$. Pilśniak and Woźniak conjectured $\chi _{nsd}^{''}(G)\le \Delta (G)+3$ for any simple graph with maximum degree $\Delta (G)$. In this paper, we prove that this conjecture holds for any planar graph with maximum degree at least 13.