A note on Goldberg's conjecture on total chromatic numbers

Abstract

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