A note on Goldberg's conjecture on total chromatic numbers

TL;DR

This paper explores Goldberg's conjecture on total chromatic numbers, showing that under certain conditions, the total chromatic number equals the chromatic index, assuming the Goldberg-Seymour conjecture.

Contribution

It demonstrates that if the chromatic index is sufficiently large, then the total chromatic number equals the chromatic index, assuming the Goldberg-Seymour conjecture.

Findings

$ ext{Total chromatic number} = ext{Chromatic index}$ if $ ext{chromatic index} ext{ is large}$

Results depend on the Goldberg-Seymour conjecture

Provides conditions for equality of total chromatic number and chromatic index

Abstract

Let $G=(V(G), E(G))$ be a multigraph with maximum degree $\Delta(G)$, chromatic index $\chi'(G)$ and total chromatic number $\chi''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $\chi''(G)\leq \Delta(G)+\mu(G) +1$ for a multigraph $G$, where $\mu(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $\chi''(G)=\chi'(G)$ if $\chi'(G)\geq \Delta(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $\chi''(G)=\chi'(G)$ if $\chi'(G)\geq \max\{ \Delta(G)+2, |V(G)|+1\}$ in this note. Consequently, $\chi''(G) = \chi'(G)$ if $\chi'(G) \ge \Delta(G) +2$ and $G$ has a spanning edge-chromatic critical subgraph.