Enumerating independent sets in Abelian Cayley graphs

We show that any connected Cayley graph Γ on an Abelian group of order 2n and degree Ω̃(log n) has at most 2(1+ o(1)) independent sets. This bound is tight up to to the o(1) term when Γ is bipartite. Our proof is based on Sapozhenko’s graph container method and uses the Plünnecke-Rusza-Petridis inequality from additive combinatorics.

[1]  Peter Keevash,et al.  Homomorphisms from the torus , 2020, 2009.08315.

[2]  J. Kahn,et al.  The Number of Maximal Independent Sets in the Hamming Cube , 2019, Comb..

[3]  Wojciech Samotij,et al.  THE METHOD OF HYPERGRAPH CONTAINERS , 2018, Proceedings of the International Congress of Mathematicians (ICM 2018).

[4]  Alfred Geroldinger,et al.  Combinatorial Number Theory and Additive Group Theory , 2009 .

[5]  Alessandro Panconesi,et al.  Concentration of Measure for the Analysis of Randomized Algorithms , 2009 .

[6]  Frank Harary,et al.  Bigraphs versus digraphs via matrices , 1980, J. Graph Theory.

[7]  Dana Randall,et al.  Phase coexistence and torpid mixing in the 3-coloring model on ℤd , 2012, SIAM J. Discret. Math..

[8]  Prasad Tetali,et al.  Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs , 2006, Random Struct. Algorithms.

[9]  Noga Alon,et al.  An Application of Graph Theory to Additive Number Theory , 1985, Eur. J. Comb..

[10]  Will Perkins,et al.  Approximately counting independent sets in bipartite graphs via graph containers , 2021, SODA.

[11]  Will Perkins,et al.  Independent sets of a given size and structure in the hypercube , 2022, Combinatorics, Probability and Computing.

[12]  Igal Sason.  A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs , 2021, Entropy.

[13]  David J. Galvin,et al.  On homomorphisms from the Hamming cube to Z , 2003 .

[14]  S. K Stein,et al.  Two Combinatorial Covering Theorems , 1974, J. Comb. Theory, Ser. A.

[15]  Jinyoung Park,et al.  Note on the Number of Balanced Independent Sets in the Hamming Cube , 2021, Electron. J. Comb..

[16]  J. Kahn,et al.  The number of 4-colorings of the Hamming cube , 2018, Israel Journal of Mathematics.

[17]  Adam Zsolt Wagner,et al.  Intersecting families of sets are typically trivial , 2021, J. Comb. Theory B.

[18]  J'ozsef Balogh,et al.  Independent sets in the middle two layers of Boolean lattice , 2020, J. Comb. Theory, Ser. A.

[19]  N. Alon.  Independent sets in regular graphs and sum-free subsets of finite groups , 1991 .

[20]  Will Perkins,et al.  Independent sets in the hypercube revisited , 2019 .

[21]  Helmut Plünnecke,et al.  Eine zahlentheoretische Anwendung der Graphentheorie. , 1970 .

[22]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[23]  Yufei Zhao,et al.  The Number of Independent Sets in a Graph with Small Maximum Degree , 2011, Graphs Comb..

[24]  Wojciech Samotij,et al.  Counting independent sets in graphs , 2014, Eur. J. Comb..

[25]  Ashwin Sah,et al.  The Number of Independent Sets in a Regular Graph , 2009, Combinatorics, Probability and Computing.

[26]  John E. Olson.  On the sum of two sets in a group , 1984 .

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

[28]  G. Petridis.  The Plünnecke–Ruzsa Inequality: An Overview , 2014 .

[29]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[30]  Daniel J. Kleitman,et al.  On the number of graphs without 4-cycles , 1982, Discret. Math..

[31]  David Galvin.  A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube , 2011, Comb. Probab. Comput..

[32]  David J. Galvin.  Independent sets in the discrete hypercube , 2019, 1901.01991.