# Beating the 2Δ bound for approximately counting colourings: a computer-assisted proof of rapid mixing

@inproceedings{Bubley1998BeatingT2, title={Beating the 2$\Delta$ bound for approximately counting colourings: a computer-assisted proof of rapid mixing}, author={Russ Bubley and Martin E. Dyer and Catherine S. Greenhill}, booktitle={SODA '98}, year={1998} }

We consider random walks on graph colourings of an nvertex graph. It has been shown by Jerrum and by Salas and Sokal that a simple random walk would mix rapidly provided the number of colours, k, exceeded the maximum degree A of the graph by a factor of at least 2. Lack of improvements on this bound led to a conjecture that k 2 2A was a natural barrier. We disprove this conjecture in the simplest case of 5-colouring graphs of maximum degree 3. Our proof involves a novel computer-assisted proof…

## 39 Citations

A more rapidly mixing Markov chain for graph colorings

- MathematicsRandom Struct. Algorithms
- 1998

A new Markov chain is defined on k-colourings of graphs, and its convergence properties are related to the maximum degree ∆ of the graph, and it is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/Salas–Sokal chain in most circumstances.

University of Birmingham Improved bounds for randomly sampling colorings via linear programming

- Mathematics
- 2018

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k -colorings of a graph G on n vertices with maximum degree ∆ is rapidly mixing for k ≥ ∆ +…

Very rapid mixing of the Glauber dynamics for proper colorings on bounded‐degree graphs

- MathematicsRandom Struct. Algorithms
- 2002

This paper extends both results to prove that the Glauber dynamics has optimal mixing time when the graph has maximum degree Δ and the number of colors is a small constant fraction smaller than 2Δ.

Vigoda's Improvement On Sampling Colorings

- Mathematics
- 2011

- In 1995, Jerrum proved a rapid mixing property on a certain Markov chain to solve the problem of sampling uniformly at random from the set of all proper k-colorings on a graph with maximum degree .…

An Extension of Path Coupling and Its Application to the Glauber Dynamics for Graph Colorings

- MathematicsSIAM J. Comput.
- 2000

It is shown that the Glauber dynamics has O(n log(n) mixing time for triangle-free $\Delta$-regular graphs if k colors are used, where $k\geq (2-\eta)\Delta$, for some small positive constant $\eta$.

On Sampling Colorings of Bipartite Graphs

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2006

It is shown that a class of markov chains cannot be used as efficient samplers and these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartites subgraph.

Improved Bounds for Randomly Sampling Colorings via Linear Programming

- MathematicsSODA
- 2019

Two approaches are used to give two proofs that the Glauber dynamics is rapidly mixing for any $k\ge\left(\frac{11}{6} - \epsilon_0\right)\Delta$ for some absolute constant $k > 2 \Delta$.

Linear Programming Bounds for Randomly Sampling

- Mathematics
- 2018

Here we study the problem of sampling random proper colorings of a bounded degree graph. Let k be the number of colors and let d be the maximum degree. In 1999, Vigoda [Vig99] showed that the Glauber…

Linear Programming Bounds for Randomly Sampling Colorings

- MathematicsArXiv
- 2018

It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive, even for the flip dynamics, and this is the first improvement to Vigoda's analysis that holds for general graphs.

An extension of path coupling and its application to the Glauber dynamics for graph colourings (extended abstract)

- MathematicsSODA '00
- 2000

A new method for analysing the mixing time of Maxkov cbain and it is shown that the Glanber dynamics has O(nlog(n) mixing time for triangle-free Aregular graphs ff k colours axe used, for some small positive constant T}.

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