Neighbor product distinguishing total colorings of corona of subcubic graphs

TL;DR

This paper investigates a special total coloring of graphs called neighbor product distinguishing total coloring, focusing on corona products of subcubic graphs, and proves an upper bound related to the maximum degree.

Contribution

It extends the concept of neighbor product distinguishing total coloring to corona products of subcubic graphs and establishes an upper bound of ext{Δ(G⊙H)}+3 colors for such colorings.

Findings

Proves an upper bound of Δ(G⊙H)+3 for the neighbor product distinguishing total chromatic number.

Extends the conjecture to corona product graphs.

Provides new insights into coloring properties of corona 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 $c(x)\neq c(y)$ 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 $\prod(u)\neq \prod(v)$, 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 $\chi"_{\prod}(G\circ H)\leq \Delta(G\circ H)+3$.