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 \(\phi\) of a graph \(G\) is called a neighbor sum distinguishing \(k\)-total coloring if \(f(u) \neq f(v)\) 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 \(\mathrm{mad}(G)\). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if \(G\) is a graph with \(\Delta(G) \geq 6\) and \(\mathrm{mad}(G) < \frac{18}{5}\), then \(\chi''_{\sum}(G) \leq \Delta(G) + 2\). This bound is sharp.