The total chromatic number $χ_{T}(G)$ of a graph $G$ is the least number $k$ such that $G$ has a total coloring with $k$ colors. The total coloring conjecture claims that $Δ(G)+1\leq χ_{T}(G)\leq Δ(G)+2$ for any graph $G$. A vertex $v$ of $G$ is said to be a major vertex if $d_{G}(v)=Δ(G)$. The author shows that $χ_{T}(G)=5$ for any graph $G$ with a unique major vertex of degree 4.
Reviewer: I.Tomescu (Bucureşti)