  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.
