# The Glauber dynamics for edge-colourings of trees

## TL;DR

This paper proves that the Glauber dynamics for edge-colourings of trees with sufficiently many colours mixes rapidly, using recursive decomposition and a new monotonicity result, establishing optimal bounds on the number of colours needed.

## Contribution

It introduces a recursive decomposition approach and a monotonicity result for Glauber dynamics, providing the first polynomial mixing time bound for edge-colourings of trees with optimal colour bounds.

## Key findings

- Glauber dynamics mixes in polynomial time for k ≥ Δ+1.
- The bound on the number of colours is optimal and necessary.
- A new monotonicity property simplifies the analysis.

## 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.