Adjacent vertex distinguishing total coloring of 3-degenerate graphs

Abstract

A total coloring of a simple undirected graph $G$ is an assignment of colors to its vertices and edges such that the colors given to the vertices form a proper vertex coloring, the colors given to the edges form a proper edge coloring, and the color of every edge is different from that of its two endpoints. That is, $\phi:V(G)\cup E(G)\rightarrow\mathbb{N}$ is a total coloring of $G$ if $\phi(u)\neq\phi(v)$ and $\phi(uv)\neq\phi(u)$ for all $uv\in E(G)$, and $\phi(uv)\neq\phi(uw)$ for any $u \in V(G)$ and distinct $v,w \in N(u)$ (here, $N(u)$ denotes the set of neighbours of $u$). A total coloring $\phi$ of a graph $G$ is said to be ``Adjacent Vertex Distinguishing'' (or AVD for short) if for all $uv\in E(G)$, we have that $\phi(\{u\}\cup\{uw:w\in N(u)\})\neq\phi(\{v\}\cup\{vw\colon w\in N(v)\})$. The AVD Total Coloring Conjecture of Zhang, Chen, Li, Yao, Lu, and Wang (Science in China Series A: Mathematics, 48(3):289--299, 2005) states that every graph $G$ has an AVD total coloring using at most $\Delta(G)+3$ colors, where $\Delta(G)$ denotes the maximum degree of $G$. For some $s\in\mathbb{N}$, a graph $G$ is said to be $s$-degenerate if every subgraph of $G$ has minimum degree at most $s$. Miao, Shi, Hu, and Luo (Discrete Mathematics, 339(10):2446--2449, 2016) showed that the AVD Total Coloring Conjecture is true for 2-degenerate graphs. We verify the conjecture for 3-degenerate graphs.