Matchings and independent sets of a fixed size in regular graphs

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 [email protected]?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.

[1]  Jeff Kahn,et al.  An Entropy Approach to the Hard-Core Model on Bipartite Graphs , 2001, Combinatorics, Probability and Computing.

[2]  Prasad Tetali,et al.  On Weighted Graph Homomorphisms , 2001, Graphs, Morphisms and Statistical Physics.

[3]  Mokshay M. Madiman,et al.  Information Inequalities for Joint Distributions, With Interpretations and Applications , 2008, IEEE Transactions on Information Theory.

[4]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[5]  Ian M. Wanless.  A lower bound on the maximum permanent in Λnk , 2003 .

[6]  Ehud Friedgut,et al.  Hypergraphs, Entropy, and Inequalities , 2004, Am. Math. Mon..

[7]  P. H. Lundow,et al.  On the Validations of the Asymptotic Matching Conjectures , 2006, math/0603001.

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

[9]  N. Alon,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2004 .

[10]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[11]  Klas Markström,et al.  On the Number of Matchings in Regular Graphs , 2008, Electron. J. Comb..

[12]  O. J. Heilmann,et al.  Theory of monomer-dimer systems , 1972 .