Total coloring graphs with large maximum degree

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