Extension from Precoloured Sets of Edges
  We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability. 
Katherine Edwards, Ant\'onio Gir\~ao, Jan van den Heuvel, Ross J.   Kang, Gregory J. Puleo, Jean-S\'ebastien Sereni
math.CO