Neighbor sum distinguishing total choosability of planar graphs

Abstract

A total-k-coloring of a graph G is a mapping $c: V(G)\cup E(G)\rightarrow \{1, 2,\dots , k\}$ such that any two adjacent or incident elements in $V(G)\cup E(G)$ receive different colors. For a total-k-coloring of G, let $\sum _c(v)$ denote the total sum of colors of the edges incident with v and the color of v. If for each edge $uv\in E(G)$, $\sum _c(u)\ne \sum _c(v)$, then we call such a total-k-coloring neighbor sum distinguishing. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by $\chi _{\Sigma }^{''}(G)$. Pilśniak and Woźniak conjectured $\chi _{\Sigma }^{''}(G)\le \Delta (G)+3$ for any simple graph with maximum degree $\Delta (G)$. In this paper, we prove that for any planar graph G with maximum degree $\Delta (G)$, $ch^{''}_{\Sigma }(G)\le \max \{\Delta (G)+3,16\}$, where $ch^{''}_{\Sigma }(G)$ is the neighbor sum distinguishing total choosability of G.

Appendix

1.1 The codes