Adjacent vertex distinguishing total chromatic number of graph products

The adjacent vertex distinguishing (AVD)-total chromatic number ${\chi }_a^{\mathrm{\prime }\mathrm{\prime }}(G)$ of a graph $G$ is the least integer $k$ for which $G$ has a proper total coloring $f$ with $k$ colors such that $C_G(u)\mathrm{\ne }{C}_G(v)$ 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 ${\chi }_a^{\mathrm{\prime }\mathrm{\prime }}(G)\le \mathrm{\Delta }(G)+3$ for every simple graph $G$, where $\mathrm{\Delta }(G)$ 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.