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