Total coloring graphs with large maximum degree

We prove that for any graph G, the total chromatic number of G is at most (G) +2 |V (G) | (G) +1. This saves one color in comparison with a result of Hind from 1992. In particular, our result says that if (G) 12|V (G) |, then G has a total coloring using at most (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< <1, there exists n₀ N such that: if G is an r-regular graph on n n₀ vertices with r 12 (1+) n, then T (G) (G) +2. This confirms the Total Coloring Conjecture for such graphs G.