A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Abstract

Let $G=(V,E)$ be a graph and $\phi$ be a total $k$-coloring of $G$ using the color set $\{1,\ldots , k\}$. Let $\sum _\phi (u)$ denote the sum of the color of the vertex $u$ and the colors of all incident edges of $u$. A $k$-neighbor sum distinguishing total coloring of $G$ is a total $k$-coloring of $G$ such that for each edge $uv\in E(G)$, $\sum _\phi (u)\ne \sum _\phi (v)$. By $\chi ^{''}_{nsd}(G)$, we denote the smallest value $k$ in such a coloring of $G$. Pilśniak and Woźniak first introduced this coloring and conjectured that $\chi _{nsd}^{''}(G)\le \Delta (G)+3$ for any simple graph $G$. In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with $\Delta (G)\ge 7$. Moreover, we also show that $\chi _{nsd}^{''}(G)\le \Delta (G)+2$ for planar graphs without intersecting triangles with $\Delta (G) \ge 9$. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.