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$c(x)\neq c(y)$ where $x$, $y\in V(G)\bigcup E(G)$ and $x$ isadjacent 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 $\prod(u)\neq \prod(v)$, then the coloring $c$ is called a neighbor product distinguishing total coloring of $G$. By $\chi''_{\prod}(G)$, we denote the minimalvalue of $k$ in such a coloring of $G$. In this paper, we considered the corona graph $G$ of two arbitrary subcubic graphs and confirmed that $\chi''_{\prod}(G)\leq\Delta(G)+3$.

Keywords

• total coloring
• corona graph
• neighbor product distinguishing coloring

Supporting Institution

School of Mathematics and Statistics

Project Number

NSFC12001332

Ethical Statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Thanks

This work was supported by the National Natural Science Foundation of China (Grant No. NSFC12001332).

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