Aspects of edge list-colourings Original Research Article

An assignment of colours to the edges of a multigraph is called an s-improper edge-colouring if no colour appears on more than s edges incident with any given vertex. We prove that if L : E(G)→2N is an assignment of lists of colours to the edges of a multigraph G with |L(e)|⩾⌈max{d(u),d(v)}/s⌉ for every edge e joining vertices u and v, and either s is even or G is bipartite, then G has an s-improper L-edge-colouring in which no colour appears on too many parallel edges. We prove these results using a new vertex-splitting lemma which generalizes the vertex-splitting lemma of Hilton et al. (J. Combin. Theory Ser. B 72 (1998) 91–103). We present applications of our results to school timetabling and conference scheduling.