A note on Goldbergs conjecture on total chromatic numbers

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.