Total Coloring Graphs With Large Maximum Degree

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta \left(G\right)+2\left(\frac{\left|V\left(G\right)\right|}{\Delta \left(G\right)+1}\right)$. This saves one color in comparison with the result of Hind from 1992. In particular, our result says that if $\Delta \left(G\right)\ge \frac{1}{2}\left|V\left(G\right)\right|$, then $G$ has a total coloring using at most $\Delta \left(G\right)+4$ colors. When $G$ is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any $0<\epsilon <1$, there exists $n_0\in N$ such that: if $G$ is an $r$-regular graph on $n\ge n_0$ vertices with $r\ge \frac{1}{2}\left(1+\epsilon \right)n$, then ${\chi }_T\left(G\right)\le \Delta \left(G\right)+2$. This confirms the Total Coloring Conjecture for such graphs $G$.