Adjacent vertex distinguishing total chromatic number of graph products

Abstract: The adjacent vertex distinguishing (AVD)-total chromatic number $χ&#39;&#39;<em>{a}(G)$ 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 $χ&#39;&#39;</em>{a}(G)\leq Δ(G)+3$ 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.