Adjacent vertex distinguishing total chromatic number of graph products

Source abstract

The adjacent vertex distinguishing (AVD)-total chromatic number $χ''_{a}(G)$ of a graph $G$ is the least integer $k$ for which $G$ has a proper total coloring $f$ with $k$ colors such that $C_G(u)\neq C_G(v)$ for every edge $uv\in E(G)$, where $C_G(u)=\{f(u)\}\cup\{f(uw):uw\in E(G)\}$. The AVD-total coloring conjecture (AVD-TCC) asserts that $χ''_{a}(G)\leq Δ(G)+3$ for every simple graph $G$, where $Δ(G)$ is the maximum degree of $G$. In this paper, we prove the AVD-TCC for certain classes of graph products, including Cartesian products, lexicographic products, skew products, cover products, comb products, and Indu--Bala products.