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A graph G with maximum degree @D and edge chromatic number @g^'(G)>@D is [email protected] if @g^'(G-e)[email protected] for every edge e of G. New lower bounds are given for the average degree of an [email protected] graph, which improve on the best bounds previously known for most values of @D. Examples of [email protected] graphs are also given. In almost all cases, there remains a large gap between the best lower bound known and the smallest average degree of any known [email protected] graph.

Let G be a simple graph with average degree d̄ and maximum degree Δ. It is proved, in this paper that G is not critical if d̄⩽6 and Δ⩾8, or d̄⩽203 and Δ⩾9. This result generalizes earlier results of Vizing (Metody Diskret. Analiz. 5 (1965) 9), Mel'nikov (Mat. Zametki 7 (1970) 671) and Hind and Zhao (Discrete Math. 190 (1998) 107) and Yan and Zhao (Graphs Combin. 16 (2) (2000) 245). It also improves a result by Fiorini (Math. Proc. Cambridge Philos. Soc. 77 (1975) 475) on the number of edges of critical graphs for certain Δ.

Vizing's adjacency lemma describes an important property of edge-chromatic critical graphs. In an edge-chromatic critical simple graph G of maximum degree ?, if xy is an edge then x is adjacent with at least ? - deg(y) + 1 vertices (?y) of degree ?. In this paper, we obtain an extension of this result to multigraphs. ? 1999 Elsevier Science B.V. All rights reserved.