# On Approximately Counting Colorings of Small Degree Graphs

@article{Bubley1999OnAC,
title={On Approximately Counting Colorings of Small Degree Graphs},
author={Russ Bubley and Martin E. Dyer and Catherine S. Greenhill and Mark Jerrum},
journal={SIAM J. Comput.},
year={1999},
volume={29},
pages={387-400}
}
• Published 1 October 1999
• Mathematics
• SIAM J. Comput.
We consider approximate counting of colorings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree $\Delta$ of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by considering the simplest case of 5-coloring graphs of maximum degree 3. Our proof involves a computer…

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## References

SHOWING 1-10 OF 23 REFERENCES
A more rapidly mixing Markov chain for graph colorings
• Mathematics
Random 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.
On Markov Chains for Independent Sets
• Mathematics
J. Algorithms
• 2000
A new rapidly mixing Markov chain for independent sets is defined and a polynomial upper bound for the mixing time of the new chain is obtained for a certain range of values of the parameter ?, which is wider than the range for which the mixingTime of the Luby?Vigoda chain is known to be polynomially bounded.
Path coupling: A technique for proving rapid mixing in Markov chains
• Mathematics
Proceedings 38th Annual Symposium on Foundations of Computer Science
• 1997
A new approach to the coupling technique, which is called path coupling, for bounding mixing rates, is illustrated, which may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method.
Approximating the Permanent
• Mathematics
SIAM J. Comput.
• 1989
A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...
Some #P-completeness Proofs for Colourings and Independent Sets
• Mathematics
• 1997
We consider certain counting problems involving colourings of graphs and independent sets in hypergraphs. Using polynomial interpolation techniques, we show that these problems are #P -complete.
On the computational complexity of the Jones and Tutte polynomials
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 1990
Abstract We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the
A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph
• M. Jerrum
• Mathematics
Random Struct. Algorithms
• 1995
A fully polynomial randomized approximation scheme is presented for estimating the number of (vertex) k‐colorings of a graph of maximum degree Δ, when k ≥ 2Δ + 1. © 1995 John Wiley & Sons, Inc.
Random Generation of Combinatorial Structures from a Uniform Distribution
• Computer Science, Mathematics
Theor. Comput. Sci.
• 1986
A random polynomial-time algorithm for approximating the volume of convex bodies
• Mathematics
JACM
• 1989
The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within K within Euclidean space.
The Markov chain Monte Carlo method: an approach to approximate counting and integration
• Computer Science
• 1996
The introduction of analytical tools with the aim of permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization.