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arXiv · 1611.09002

A precolouring extension of Vizing's theorem

António Girão·Ross J. Kang
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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 $5$.

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António Girão, Ross J. Kang. 2016-11-28. A precolouring extension of Vizing's theorem. https://arxiv.org/abs/1611.09002

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