Over on the wikipedia article on the total chormatic number $\chi^{''}$ I found the following sentence:
$\ldots$ if the list coloring conjecture is true, then $\chi^{''}(G) \le \Delta (G) + 3$
I do not see why this should be the case, could you help me?
Let $C$ be a set of colors, $|C|=\Delta(G)+3$. Since $\chi(G)\le\Delta(G)+1\lt|C|$, there is a proper vertex coloring $f:V(G)\to C$. To each edge $e=uv\in E(G)$ assign the list $L_e=C\setminus\{f(u),f(v)\}$. By the list coloring conjecture and Vizing's theorem, $\operatorname{ch}'(G)=\chi'(G)\le\Delta(G)+1$. Since $|L_e|=\Delta(G)+1$ for each edge $e$, there is a proper edge coloring $g:E(G)\to C$ such that $g(e)\in L_e$ for each edge $e\in E(G)$. Now $f\cup g:V(G)\cup E(G)\to C$ is a proper total coloring, showing that $\chi''(G)\le\Delta(G)+3$.