The Total Chromatic Number of Graphs of High Minimum Degree
Amanda G. Chetwynd, A. J. W. Hilton, Zhao Cheng · Journal of the London Mathematical Society · 1991
If G is a simple graph with minimum degree δ(G) satisfying δ(G)⩾ ⅚ f(|V(G|+1) the total chromatic number conjecture holds; moreover if δ(G) ⩾ ¾ |V(G| then χT(G) ⩽ Δ(G)+3. Also if G has odd order and is regular with d{G) ⩾ ⅓√7|(G)| then a necessary and sufficient condition for χT(G) = Δ(G)+1 is given.