Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

Abstract This survey concerns regular graphs that are extremal with respect to the number of independent sets and, more generally, graph homomorphisms. More precisely, in the family of of d-regular graphs, which graph G maximizes/minimizes the quantity i(G)1/v(G), the number of independent sets in G normalized exponentially by the size of G? What if i (G) is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.

[1]  H. Whitney,et al.  An inequality related to the isoperimetric inequality , 1949 .

[2]  Henryk Minc,et al.  Upper bounds for permanents of $\left( {0,\,1} \right)$-matrices , 1963 .

[3]  E. Lieb,et al.  Best Constants in Young's Inequality, Its Converse, and Its Generalization to More than Three Functions , 1976 .

[4]  M. Voorhoeve,et al.  A lower bound for the permanents of certain (0,1)-matrices , 1978 .

[5]  Norbert Seifter,et al.  Upper bounds for permanents of (1, − 1)-matrices , 1984 .

[6]  Fan Chung Graham,et al.  Some intersection theorems for ordered sets and graphs , 1986, J. Comb. Theory, Ser. A.

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

[8]  Helmut Finner,et al.  A Generalization of Holder's Inequality and Some Probability Inequalities , 1992 .

[9]  Alexander Schrijver,et al.  Counting 1-Factors in Regular Bipartite Graphs , 1998, J. Comb. Theory B.

[10]  Graham R. Brightwell,et al.  Graph Homomorphisms and Phase Transitions , 1999, J. Comb. Theory, Ser. B.

[11]  Olle Häggström,et al.  Nonmonotonic Behavior in Hard-Core and Widom–Rowlinson Models , 1999 .

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

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

[14]  J. Kahn.  Entropy, independent sets and antichains: A new approach to Dedekind's problem , 2001 .

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

[16]  David J. Galvin.  Bounding the Partition Function of Spin-Systems , 2006, Electron. J. Comb..

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

[18]  David Gamarnik,et al.  Counting without sampling: Asymptotics of the log‐partition function for certain statistical physics models , 2008, Random Struct. Algorithms.

[19]  G. Katona.  A theorem of finite sets , 2009 .

[20]  The Number of Independent Sets in a Regular Graph , 2009, Combinatorics, Probability and Computing.

[21]  Prasad Tetali,et al.  Matchings and independent sets of a fixed size in regular graphs , 2009, J. Comb. Theory, Ser. A.

[22]  Alexander Schrijver,et al.  On Leonid Gurvits’s Proof for Permanents , 2010, Am. Math. Mon..

[23]  O. Pikhurko,et al.  Maximizing the number of q‐colorings , 2008, 0811.2625.

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

[25]  Jonathan Cutler,et al.  Extremal graphs for homomorphisms , 2011, J. Graph Theory.

[26]  Leonid Gurvits,et al.  Unleashing the power of Schrijver's permanental inequality with the help of the Bethe Approximation , 2011, Electron. Colloquium Comput. Complex..

[27]  David Galvin.  Two problems on independent sets in graphs , 2011, Discret. Math..

[28]  Yufei Zhao,et al.  The Bipartite Swapping Trick on Graph Homomorphisms , 2011, SIAM J. Discret. Math..

[29]  Jonathan Cutler,et al.  Independent Sets in Graphs with Given Minimum Degree , 2012, Electron. J. Comb..

[30]  David Galvin,et al.  Maximizing H‐Colorings of a Regular Graph , 2011, J. Graph Theory.

[31]  Jonathan Cutler,et al.  Coloring Graphs with Graphs : a Survey , 2013 .

[32]  Jonathan Cutler,et al.  The maximum number of complete subgraphs in a graph with given maximum degree , 2014, J. Comb. Theory, Ser. B.

[33]  P. Csikvári.  Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems , 2014, 1406.0766.

[34]  Jonathan Cutler,et al.  Extremal Graphs for Homomorphisms II , 2014, J. Graph Theory.

[35]  David Galvin,et al.  Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree , 2014, J. Graph Theory.

[36]  David J. Galvin,et al.  Three tutorial lectures on entropy and counting , 2014, 1406.7872.

[37]  Allan Sly,et al.  Counting in two-spin models on d-regular graphs , 2014 .

[38]  Benny Sudakov,et al.  Maximizing the Number of Independent Sets of a Fixed Size , 2015, Comb. Probab. Comput..

[39]  Jie Ma,et al.  Maximizing proper colorings on graphs , 2014, J. Comb. Theory, Ser. B.

[40]  John Engbers,et al.  Extremal H‐Colorings of Graphs with Fixed Minimum Degree , 2013, J. Graph Theory.

[41]  Yufei Zhao,et al.  On replica symmetry of large deviations in random graphs , 2012, Random Struct. Algorithms.

[42]  P'eter Csikv'ari.  Extremal regular graphs: the case of the infinite regular tree , 2016, 1612.01295.

[43]  B. Roberts,et al.  On the average size of independent sets in triangle-free graphs , 2016, 1606.01043.

[44]  Will Perkins,et al.  On the Widom-Rowlinson Occupancy Fraction in Regular Graphs , 2017, Comb. Probab. Comput..

[45]  Christian Reiher,et al.  The clique density theorem , 2012, 1212.2454.

[46]  Will Perkins,et al.  Independent sets, matchings, and occupancy fractions , 2015, J. Lond. Math. Soc..

[47]  Péter Csikvári,et al.  Sidorenko's Conjecture, Colorings and Independent Sets , 2017, Electron. J. Comb..

[48]  Will Perkins,et al.  The Widom-Rowlinson model, the hard-core model and the extremality of the complete graph , 2017, Eur. J. Comb..

[49]  Jonathan Cutler,et al.  Minimizing the number of independent sets in triangle-free regular graphs , 2018, Discret. Math..

[50]  Luke Sernau.  Graph operations and upper bounds on graph homomorphism counts , 2018, J. Graph Theory.

[51]  Will Perkins,et al.  Extremes of the internal energy of the Potts model on cubic graphs , 2016, Random Struct. Algorithms.

[52]  David Conlon,et al.  Sidorenko's conjecture for blow-ups , 2018 .

[53]  Will Perkins,et al.  Counting independent sets in cubic graphs of given girth , 2018, J. Comb. Theory, Ser. B.

[54]  C. Tsallis.  Entropy , 2022, Thermodynamic Weirdness.

[55]  Stefan Adams,et al.  The Widom-Rowlinson model on the Delaunay graph , 2017, Electronic Journal of Probability.