The number of independent sets in graphs

Sapozhenko, A. A.

doi:10.3103/s0027132207030072

Cited by 17 publications

(29 citation statements)

References 2 publications

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“…In major breakthroughs a little over five years ago, Conlon and Gowers [21] and independently Schacht [55] proved very general transference results, yielding important corollaries of the KŁR conjecture. Their work was soon followed by another dramatic breakthrough: Balogh, Morris, and Samotij [13] and independently Saxton and Thomason [53], building on work of Kleitman-Winston [39] and of Sapozhenko [51,52] for graphs, developed powerful theories of hypergraph containers.…”

Section: 3mentioning

confidence: 99%

Multicolor containers, extremal entropy, and counting

Falgas–Ravry

O'Connell

Uzzell

2018

Random Struct Algorithms

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In breakthrough results, Saxton‐Thomason and Balogh‐Morris‐Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton‐Thomason and an entropy‐based framework to deduce container and counting theorems for hereditary properties of k‐colorings of very general objects, which include both vertex‐ and edge‐colorings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterization and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity, and multicolored graphs among others. Similar results were recently and independently obtained by Terry.

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Section: 3mentioning

confidence: 99%

Multicolor containers, extremal entropy, and counting

Falgas–Ravry

O'Connell

Uzzell

2018

Random Struct Algorithms

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“…As many interesting combinatorial problems can be phrased in terms of independent sets in graphs and hypergraphs, the problem of bounding the number of independent sets is of central interest. For example, see the ICM 2018 survey [4] on the recent breakthroughs on the hypergraph container method of Balogh, Morris, and Samotij [3] and independently Saxton and Thomason [25], which built partly on the earlier work by Sapozhenko [24], a precursor to Theorem 1.1, giving a weaker upper bound for i(G).…”

Section: Introductionmentioning

confidence: 99%

The number of independent sets in an irregular graph

Sah

Sawhney

Stoner

et al. 2019

Journal of Combinatorial Theory, Series B

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Settling Kahn's conjecture (2001), we prove the following upper bound on the number i(G) of independent sets in a graph G without isolated vertices:where du is the degree of vertex u in G. Equality occurs when G is a disjoint union of complete bipartite graphs. The inequality was previously proved for regular graphs by Kahn and Zhao.We also prove an analogous tight lower bound:where equality occurs for G a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.

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“…The proofs of Theorems 1.3 and 1.4 are similar, while that of Theorem 1.5 is related but somewhat trickier. The general approach has its roots in an idea for counting (ordinary) independent sets due to A.A. Sapozhenko [9], [10].…”

Section: Introductionmentioning

confidence: 99%