Generalization of two results of Hilton on total-colourings of a graph

Abstract We generalize two results of Hilton on total-colourings of a graph. The first generalized result unifies several previous results/proof techniques of Bermond, Chen, Chew, Fu, Hilton, Wang, and Yap. Applying the second generalized result, we prove that if G ⊆ K n , n is such that Δ ( G ) = n − 1 and the complement of G with respect to K n , n contains a 1-factor, then its total chromatic number is Δ ( G ) + 1.