On Vizing's theorem, adjacency lemma and fan argument generalized to multigraphs

摘要

The aim of this paper is to give an alternative proof of a result in edge colourings, proved by Ehrenfeucht et al. (1984). From this result, we show that the multigraph version of Vizing's theorem, Vizing's adjacency lemma and Vizing's fan argument can be obtained immediately. The proof given here is based on fan and counting arguments involving small number of colour changes and is considerably simpler and shorter than the one in Ehrenfeucht et al. (1984).