The total coloring of $K_5$-minor-free graphs

TL;DR

This paper proves bounds on the total chromatic number of $K_5$-minor-free graphs, showing it is at most $ ext{max}( ext{deg}+2, ext{deg}+1)$ depending on the maximum degree, advancing understanding of graph coloring.

Contribution

It establishes new upper bounds for the total chromatic number of $K_5$-minor-free graphs based on maximum degree, improving previous results.

Findings

For $ ext{deg} eq 10$, $ ext{total chromatic number} oxed{ ext{≤} ext{deg} + 2}$.

For $ ext{deg} eq 7$, $ ext{total chromatic number} = ext{deg} + 1$ when $ ext{deg} ext{≥} 10$.

The bounds depend on the maximum degree, refining the total coloring conjecture for this class of graphs.

Abstract

A total $k$-coloring of a graph $G$ is a coloring of $V(G)\cup E(G)$ using $k$ colors such that no two adjacent or incident elements receive the same color. The total chromatic number $\chi"(G)$ of $G$ is the smallest integer $k$ such that $G$ has a total $k$-coloring. In the paper, it is proved that for any $K_5$-minor-free graph $G$, $\chi''(G)\leq \Delta(G)+2$ if $\Delta(G)\geq 7$. Moreover, $\chi"(G)=\Delta(G)+1$ if $\Delta(G)\geq 10$.