Local version of Vizing's theorem for multi-graphs
Abstract
Extending a result of Christiansen, we prove that every mutli-graph G=(V,E) admits a proper edge colouring φ:E \1,2,…\ which is local, that is, φ(e) \d(x)+π(x),d(y)+π(y)\ for every edge e with end-points x,y∈ V, where d(z) (resp.\ π(z)) denotes the degree of a vertex z (resp.\ the maximum edge multiplicity at z). This is derived from a local version of the Fan Equation.
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