An extension of Vizing's theorem

Let G = (V (G), E (G)) be a simple graph of maximum degree δ ≤ D such that the graph induced by vertices of degree D is either a null graph or is empty. We give an upper bound on the number of colours needed to colour a subset S of V (G) ∪ E (G) such that no adjacent or incident elements of S receive the same colour. In particular, if S = E (G), we have the chromatic index χ′(G) ≤ D whereas if S = V (G) ∪ E (G) and for some positive integer k, we have the total chromatic number _χT(G) ≤ D + k. © 1997 John Wiley & Sons, Inc.