Neighbor Sum Distinguishing Total Coloring of a Kind of Sparse Graph

Abstract

For a given graph $G=(V,E)$, by $f(v)$, we denote the sum of the color on the vertex $v$ and the colors on the edges incident with $v$. A proper $k$-total coloring $\varphi$ of a graph $G$ is called a neighbor sum distinguishing $k$-total coloring if $f(u)\ne f(v)$ for each edge $uv\in E(G)$. The smallest number $k$ in such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $\chi {”}_{\sum }(G)$. The maximum average degree of $G$ is the maximum of the average degree of its non-empty subgraphs, which is denoted by $\mathrm{m}\mathrm{a}\mathrm{d}(G)$. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if $G$ is a graph with $\mathrm{\Delta }(G)\ge 6$ and $\mathrm{m}\mathrm{a}\mathrm{d}(G)<\frac{18}{5}$, then ${\chi }_{\sum }^{″}(G)\le \mathrm{\Delta }(G)+2$. This bound is sharp.