ADJACENT VERTEX DISTINGUISHING TOTAL COLORING OF GRAPHS WITH LOWER AVERAGE DEGREE

Abstract

An adjacent vertex distinguishing total coloring of a graph $G$ is a proper total coloring of $G$ such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of $G$ is denoted by $\chi''_{a}(G)$. Let mad$(G)$ and $\Delta(G)$ denote the maximum average degree and the maximum degree of a graph $G$, respectively. In this paper, we prove the following results: (1) If $G$ is a graph with mad$(G)\lt 3$ and $\Delta(G)\ge 5$, then $\Delta(G)+1\le \chi''_{a}(G)\le \Delta(G)+2$, and $\chi''_{a}(G)=\Delta(G)+2$ if and only if $G$ contains two adjacent vertices of maximum degree; (2) If $G$ is a graph with mad$(G)\lt 3$ and $\Delta(G)\le 4$, then $\chi''_{a}(G)\le 6$; (3) If $G$ is a graph with mad$(G)\lt \frac 83$ and $\Delta(G)\le 3$, then $\chi''_{a}(G)\le 5$.