A precolouring extension of Vizing's theorem

Abstract

Fix a palette $\mathcal{K}$ of $\Delta +1$ colours, a graph with maximum degree $\Delta$, and a subset $M$ of the edge set with minimum distance between edges at least $9$. If the edges of $M$ are arbitrarily precoloured from $\mathcal{K}$, then there is guaranteed to be a proper edge-colouring using only colours from $\mathcal{K}$ that extends the precolouring on $M$ to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the distance can be lowered to $5$ when the graph contains no cycle of length $5$.

arXiv comment6 pages; v2 accepted to Journal of Graph Theory