Glauber dynamics for colourings of chordal graphs and graphs of bounded treewidth

TL;DR

This paper proves that the Glauber dynamics for graph colourings mixes in polynomial time on chordal graphs and graphs of bounded treewidth under certain colour constraints, advancing understanding of sampling colourings efficiently.

Contribution

It establishes polynomial mixing times for Glauber dynamics on chordal graphs and bounded treewidth graphs with specific numbers of colours, extending previous results.

Findings

Polynomial mixing time on graphs with bounded treewidth and ≥ Δ+2 colours.

Polynomial mixing time on chordal graphs with ≥ (1+ε)(Δ+1) colours.

Supports conjecture of polynomial mixing time for certain graph classes.

Abstract

The Glauber dynamics on the colourings of a graph is a random process which consists in recolouring at each step a random vertex of a graph with a new colour chosen uniformly at random among the colours not already present in its neighbourhood. It is known that when the total number of colours available is at least $\Delta +2$, where $\Delta$ is the maximum degree of the graph, this process converges to a uniform distribution on the set of all the colourings. Moreover, a well known conjecture is that the time it takes for the convergence to happen, called the mixing time, is polynomial in the size of the graph. Many weaker variants of this conjecture have been studied in the literature by allowing either more colours, or restricting the graphs to particular classes, or both. This paper follows this line of research by studying the mixing time of the Glauber dynamics on chordal graphs, as well as graphs of bounded treewidth. We show that the mixing time is polynomial in the size of the graph in the two following cases: - on graphs with bounded treewidth, and at least $\Delta +2$ colours, - on chordal graphs if the number of colours is at least $(1+\varepsilon) (\Delta +1)$, for any fixed constant $\varepsilon$.