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Keywords:

  • multigraph;
  • edge colouring;
  • Vizing's theorem

Abstract

We consider multigraphs G for which equality holds in Vizing's classical edge colouring bound χ′(G)≤Δ + µ, where Δ denotes the maximum degree and µ denotes the maximum edge multiplicity of G. We show that if µ is bounded below by a logarithmic function of Δ, then G attains Vizing's bound if and only if there exists an odd subset SV(G) with |S|≥3, such that |E[S]|>((|S| − 1)/2)(Δ + µ − 1). The famous Goldberg–Seymour conjecture states that this should hold for all µ≥2. We also prove a similar result concerning the edge colouring bound χ′(G)≤Δ + ⌈µ/⌊g/2⌋⌉, due to Steffen (here g denotes the girth of the underlying graph). Finally we give a general approximation towards the Goldberg-Seymour conjecture in terms of Δ and µ. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:160-168, 2012