Total coloring graphs with large maximum degree

TL;DR

This paper establishes new upper bounds on the total chromatic number of graphs with large maximum degree, improving previous results and confirming the Total Coloring Conjecture for certain regular graphs.

Contribution

It provides a tighter upper bound on total coloring numbers for graphs with large maximum degree, especially for regular graphs, advancing the understanding of the Total Coloring Conjecture.

Findings

For any graph, total chromatic number ≤ Δ(G)+2⌈|V(G)|/(Δ(G)+1)⌉.

If Δ(G) ≥ ½|V(G)|, then total chromatic number ≤ Δ(G)+4.

For regular graphs with sufficiently many vertices and degree proportionally large, total chromatic number ≤ Δ(G)+2.

Abstract

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result says that if $\Delta(G)\ge \frac{1}{2}|V(G)|$, then $G$ has a total coloring using at most $\Delta(G)+4$ colors. When $G$ is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any $0<\varepsilon <1$, there exists $n_0\in \mathbb{N}$ such that: if $G$ is an $r$-regular graph on $n \ge n_0$ vertices with $r\ge \frac{1}{2}(1+\varepsilon) n$, then $\chi_T(G) \le \Delta(G)+2$. This confirms the Total Coloring Conjecture for such graphs $G$.