Note: Matchings and independent sets of a fixed size in regular graphs

Authors:
Teena Carroll;David Galvin;Prasad Tetali
Affiliations:
Mathematics Department, St. Norbert College, De Pere, WI, United States;Department of Mathematics, University of Notre Dame, South Bend, IN, United States;School of Mathematics & School of Computer Science, Georgia Institute of Technology, Atlanta, GA, United States
Venue:
Journal of Combinatorial Theory Series A
Year:
2009

Citing 2
Cited 1

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

Combinatorics, Probability and Computing
Information inequalities for joint distributions, with interpretations and applications

IEEE Transactions on Information Theory

The number of independent sets in a regular graph

Combinatorics, Probability and Computing

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Abstract

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size @? in a d-regular graph on N vertices. For 2@?N bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size @? in the graph consisting of N2d disjoint copies of K"d","d. This provides asymptotic evidence for a conjecture of S. Friedland et al. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.