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 $\varphi :E\to \{1,2,\dots \}$ which is local, that is, $\varphi (e)\le \max \{d(x)+\pi (x),d(y)+\pi (y)\}$ for every edge $e$ with end-points $x,y\in V$, where $d(z)$ (resp.\ $\pi (z)$) 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.

10 pages, minor revision