Randomly coloring planar graphs with fewer colors than the maximum degree

We study Markov chains for randomly sampling k-colorings of a graph with maximum degree Δ. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when . Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most , for fixed . The main challenge when is the possibility of “frozen” vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when , even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using “local uniformity” properties. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 731–759, 2015

[1]  Zdenek Dvorak,et al.  Spectral radius of finite and infinite planar graphs and of graphs of bounded genus , 2009, J. Comb. Theory, Ser. B.

[2]  Alistair Sinclair,et al.  Mixing in time and space for discrete spin systems , 2004 .

[3]  Dana Randall,et al.  Markov Chain Algorithms for Planar Lattice Structures , 2001, SIAM J. Comput..

[4]  L. A. Goldberg,et al.  The mixing time of Glauber dynamics for coloring regular trees , 2008 .

[5]  Eric Vigoda,et al.  Phase transition for the mixing time of the Glauber dynamics for coloring regular trees , 2009, SODA '10.

[6]  Yuval Peres,et al.  The Glauber Dynamics for Colourings of Bounded Degree Trees , 2009, APPROX-RANDOM.

[7]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[8]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .

[9]  Allan Sly,et al.  Communications in Mathematical Physics Reconstruction of Random Colourings , 2009 .

[10]  Thomas P. Hayes,et al.  Coupling with the stationary distribution and improved sampling for colorings and independent sets , 2005, SODA '05.

[11]  B. Reed.  Graph Colouring and the Probabilistic Method , 2001 .

[12]  Eric Vigoda,et al.  Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region , 2013, STOC.

[13]  P. Diaconis,et al.  COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS , 1993 .

[14]  Thomas P. Hayes.  A simple condition implying rapid mixing of single-site dynamics on spin systems , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[15]  J. Jonasson.  Uniqueness of uniform random colorings of regular trees , 2002 .

[16]  Martin E. Dyer,et al.  Randomly coloring constant degree graphs , 2004, Random Struct. Algorithms.

[17]  A. Sinclair,et al.  Fast mixing for independent sets, colorings, and other models on trees , 2007 .

[18]  Eric Vigoda,et al.  Randomly coloring sparse random graphs with fewer colors than the maximum degree , 2006 .

[19]  Thomas P. Hayes,et al.  A general lower bound for mixing of single-site dynamics on graphs , 2005 .

[20]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[21]  Alan M. Frieze,et al.  On randomly colouring locally sparse graphs , 2006, Discret. Math. Theor. Comput. Sci..

[22]  Elchanan Mossel,et al.  Gibbs rapidly samples colorings of G(n, d/n) , 2007, 0707.3241.

[23]  Leslie Ann Goldberg,et al.  Random sampling of 3-colorings in Z 2 , 2004 .

[24]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[25]  Charilaos Efthymiou.  MCMC sampling colourings and independent sets of G(n, d/n) near uniqueness threshold , 2014, SODA.

[26]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

[27]  Eric Vigoda,et al.  Reconstruction for Colorings on Trees , 2007, SIAM J. Discret. Math..

[28]  M. Jerrum.  Counting, Sampling and Integrating: Algorithms and Complexity , 2003 .

[29]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[30]  Martin E. Dyer,et al.  Randomly coloring graphs with lower bounds on girth and maximum degree , 2003, Random Struct. Algorithms.

[31]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.

[32]  M. Dyer,et al.  Mixing in time and space for lattice spin systems: A combinatorial view , 2002, International Workshop Randomization and Approximation Techniques in Computer Science.