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arXiv · 1812.05577

The Glauber dynamics for edge-colourings of trees

Michelle Delcourt·Marc Heinrich·Guillem Perarnau
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Abstract

Let $T$ be a tree on $n$ vertices and with maximum degree $\Delta$. We show that for $k\geq \Delta+1$ the Glauber dynamics for $k$-edge-colourings of $T$ mixes in polynomial time in $n$. The bound on the number of colours is best possible as the chain is not even ergodic for $k \leq \Delta$. Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees using block dynamics and chain comparison techniques. Of independent interest, we also introduce a monotonicity result for Glauber dynamics that simplifies our proof.

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Michelle Delcourt, Marc Heinrich, Guillem Perarnau. 2018-12-13. The Glauber dynamics for edge-colourings of trees. https://arxiv.org/abs/1812.05577

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