Neighbor product distinguishing total colorings of corona of subcubic graphs

Abstract

A proper $[k]$-total coloring $c$ of a graph $G$ is a mapping $c$ from $V(G)\bigcup E(G)$ to $[k]=\{1,2,\cdots ,k\}$ such that $c(x)\ne c(y)$ for which $x$, $y\in V(G)\bigcup E(G)$ and $x$ is adjacent to or incident with $y$. Let $\prod (v)$ denote the product of $c(v)$ and the colors on all the edges incident with $v$. For each edge $uv\in E(G)$, if $\prod (u)\ne \prod (v)$, then the coloring $c$ is called a neighbor product distinguishing total coloring of $G$. we use $\chi {"}_{\prod }(G)$ to denote the minimal value of $k$ in such a coloring of $G$. In 2015, Li et al. conjectured that $\Delta (G)+3$ colors enable a graph to have a neighbor product distinguishing total coloring. In this paper, we consider the neighbor product distinguishing total coloring of corona product $G\circ H$, and obtain that $\chi {"}_{\prod }(G\circ H)\le \Delta (G\circ H)+3$.