Total Coloring Graphs With Large Maximum Degree

ABSTRACT We prove that for any graph , the total chromatic number of is at most . This saves one color in comparison with the result of Hind from 1992. In particular, our result says that if , then has a total coloring using at most colors. When is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any , there exists such that: if is an ‐regular graph on vertices with , then . This confirms the Total Coloring Conjecture for such graphs .