Electronic Journal of Probability

Sampling 3-colourings of regular bipartite graphs

David Galvin

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We show that if $G=(V,E)$ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if $M$ is a Markov chain on the set of proper 3-colourings of $G$ which updates the colour of at most $c|V|$ vertices at each step and whose stationary distribution is uniform, then for $c < .22$ and $d$ sufficiently large the convergence to stationarity of $M$ is (essentially) exponential in $|V|$. In particular, if $G$ is the $d$-dimensional hypercube $Q_d$ (the graph on vertex set $\{0,1\}^d$ in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the well-known Glauber (single-site update) dynamics is exponentially slow in $2^d/(\sqrt{d} \log d )$. A combinatorial corollary of our main result is that in a uniform 3-colouring of $Q_d$ there is an exponentially small probability (in $2^d$) that there is a colour $i$ such the proportion of vertices of the even subcube coloured $i$ differs from the proportion of the odd subcube coloured $i$ by at most .22. Our proof combines a conductance argument with combinatorial enumeration methods.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 16, 481-497.

Accepted: 18 April 2007
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Mixing time 3-colouring Potts model conductance Glauber dynamics discrete hypercube

This work is licensed under aCreative Commons Attribution 3.0 License.


Galvin, David. Sampling 3-colourings of regular bipartite graphs. Electron. J. Probab. 12 (2007), paper no. 16, 481--497. doi:10.1214/EJP.v12-403. https://projecteuclid.org/euclid.ejp/1464818487

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