Counting independent sets in regular hypergraphs
CCOUNTING INDEPENDENT SETS IN REGULAR HYPERGRAPHS
J ´OZSEF BALOGH, B´ELA BOLLOB ´AS, AND BHARGAV NARAYANAN
Abstract.
Amongst d -regular r -uniform hypergraphs on n vertices, which ones have the largest numberof independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is proposesuch a generalisation. Lending credence to our conjecture, we verify it within the class of ‘quasi-bipartite’hypergraphs (a generalisation of bipartite graphs that seems natural in this context) by adopting the entropicapproach of Kahn. Introduction
This paper concerns the hypergraph analogue of an old (and now resolved) graph-theoretic problem ofGranville (see [1]). Granville raised the following problem: which d -regular graphs on n vertices have themaximum number of independent sets? This problem was also considered by Kahn [5] in the context of thehard-core model, and a complete answer is now available owing to the work of Kahn [5] and Zhao [7]: theextremal graphs are precisely those consisting of disjoint copies of the complete bipartite graph K d,d . Bynow, much more is known; see [4, 6, 8] for a small sample of the literature.Here, we shall focus on the more general problem of maximising the number of independent sets in r -uniform hypergraphs (or r -graphs, for short). While this is a natural problem, we emphasise that it is noteven apparent what the correct conjectural analogue of the complete bipartite graph is; our aim in this shortnote is to remedy this situation. A brief word about notation: a subset of the vertex set of an r -graph is independent if it induces no edges, the degree of a vertex is the number of edges containing it, and an r -graphis d -regular if each of its vertices has degree d .The following construction will play a fundamental role in our arguments. For r ≥ d ∈ N , let H rd bethe d -regular r -partite r -graph on rd vertices whose d edges are as follows: mark a subset of the vertex setof order d , partition the remaining ( r − d vertices into d sets of r − r − H rd each r -set consisting of a marked vertexand a matching edge. For example, H d is the complete bipartite graph K d,d , and H d is the set of triangles inthe graph on 3 d vertices where d vertices are each joined to both ends of all the edges of a matching coveringthe other 2 d vertices.Writing ind( G ) for the number of independent sets in an r -graph G , an easy computation tells us thatind( H rd ) = 2 ( r − d + (2 d − r − − d . Our main reason for writing this note is to make the following conjecture.
Conjecture 1.1.
For all r ≥ and d ∈ N , if G is a d -regular r -graph on n vertices , then ind( G ) ≤ ind( H rd ) n/rd . Date : 29 January 2020.2010
Mathematics Subject Classification . Primary 05A16; Secondary 05C65, 05C35. a r X i v : . [ m a t h . C O ] F e b irst, by way of orientation, let us mention that when r ≥
3, a disjoint union of copies of H rd has strictlymore independent sets than a comparable disjoint union of copies of a complete r -partite r -graph (the naturalfirst guess), and of an r -partite transversal design (the natural guess within the class of linear r -graphs);of course, strictly speaking, we mean this when the numerics allow the latter two constructions (i.e., when d = t r − for some t ∈ N for complete r -partite r -graphs, and d sufficiently large for r -partite transversaldesigns). Second, as remarked earlier, Conjecture 1.1 for r = 2 is the aforementioned Kahn–Zhao theorem,but we are unable to verify it for r ≥
3; nevertheless, in the spirit of Kahn [5], we shall verify our conjecturefor ‘quasi-bipartite’ r -graphs when r ≥ linear r -graphs, and raise the question of what one can say about regular r -graphs in general.This paper is organised as follows. In Section 2, we prove Conjecture 1.1 within the class of quasi-bipartite r -graphs, and conclude in Section 3 by discussing the main obstacles a proof of Conjecture 1.1 would have toovercome. 2. Quasi-bipartite hypergraphs
Recall that, in an r -graph G , the link L ( v ) of a vertex v is the ( r − S such that S ∪ { v } is an edge of G (and whose vertex set is precisely the span of these edges). Wesay that an r -graph G is quasi-bipartite if its vertices may be partitioned into two sets A and B such that(I) every edge of G intersects A in exactly one vertex, and(II) for each a ∈ A , the link L ( a ) of a is a matching.Two remarks about this definition are worth recording: first, when r = 2, this is easily seen to be preciselythe definition of a bipartite graph, and second, we note that H rd is quasi-bipartite for all r ≥ d ∈ N .Following Kahn [5], we prove Conjecture 1.1 for quasi-bipartite hypergraphs. Theorem 2.1. If G is a d -regular quasi-bipartite r -graph on n vertices , then ind( G ) ≤ ind( H rd ) n/rd . Proof . Denote by A ∪ B the vertex partition associated with G . Let X be the characteristic vector of arandomly chosen independent set of G . For a set S of vertices, we write X S for the subvector of X indexedby the vertices in S , and abbreviate X { v } by X v . Now, representing X = ( X A , X B ) and writing H ( X ) forthe entropy of X , we note thatlog(ind( G )) = H ( X ) = H ( X B ) + H ( X A | X B ) , and observe that { V ( L ( a )) : a ∈ A } is a d -covering of B ; this fact follows from Condition (II). Hence, byShearer’s lemma (see [2]), we have H ( X B ) ≤ d (cid:88) a ∈ A H (cid:0) X V ( L ( a )) (cid:1) . Next, note also that H ( X A | X B ) ≤ (cid:88) a ∈ A H ( X a | X B ) = (cid:88) a ∈ A H (cid:0) X a | X V ( L ( a )) (cid:1) . inally, putting these estimates together, we conclude that H ( X ) ≤ d (cid:32)(cid:88) a ∈ A H (cid:0) X V ( L ( a )) (cid:1) + d · H (cid:0) X a | X V ( L ( a )) (cid:1)(cid:33) . Now, fix an arbitrary a ∈ A and consider a set I ⊂ V ( L ( a )) that is independent in G . We write p ( I ) forthe probability that X V ( L ( a )) = I , and λ ( I ) for the number of ways that a can be added to I (which is either1 or 2) whilst preserving independence in G . In this language, we have H (cid:0) X V ( L ( a )) (cid:1) + d · H (cid:0) X a | X V ( L ( a )) (cid:1) = (cid:88) I p ( I ) (cid:18) log (cid:18) p ( I ) (cid:19) + d · H (cid:0) X a | { X V ( L ( a )) = I } (cid:1)(cid:19) . Since H ( X a | { X V ( L ( a )) = I } ) ≤ log( λ ( I )), the right hand side of the above inequality is bounded above by (cid:88) I p ( I ) (cid:18) log 1 p ( I ) + d · log( λ ( I )) (cid:19) = (cid:88) I p ( I ) log (cid:18) λ ( I ) d p ( I ) (cid:19) ≤ log (cid:32)(cid:88) I λ ( I ) d (cid:33) , where the last inequality is a consequence of Jensen’s inequality. Noting that I ranges over the subsets of V ( L ( a )) that are independent in G , we have (cid:88) I λ ( I ) d ≤ | V ( L ( a )) | + (2 d −
1) ind( L ( a )) ≤ ( r − d + (2 d − r − − d = ind( H rd ) . For the first inequality above, notice that each subset of V ( L ( a )) contributes at most 1 to the sum on the left,unless it happens to be independent in L ( a ), in which case it contributes an additional 2 d −
1. To see whythe second inequality above holds, first observe that we trivially have | V ( L ( a )) | ≤ ( r − d . Then, observethat we may bound ind( L ( a )) again using the subadditivity of entropy (the trivial case of Shearer’s lemma):writing S , S , . . . , S d for the d edges of the ( r − L ( a ), consideration of the characteristic vector Y of a randomly chosen independent set of L ( a ) leads us to conclude that H ( Y ) ≤ (cid:80) di =1 H ( Y S i ), and observingthat H ( Y S i ) ≤ log(2 r − −
1) for each 1 ≤ i ≤ d yields the required bound.Finally, putting these estimates together, and using the fact that | A | = n/r since G is d -regular, allows usto conclude that H ( X ) ≤ nrd log(ind( H rd )) , and the result follows. (cid:3) Conclusion
There seem to be two major obstacles in adapting our arguments here for quasi-bipartite r -graphs to dealwith general r -graphs; we outline these below briefly.First, while any entropic approach would seem to demand something like Condition (I), Condition (II)appears to be an artefact of our proof; this latter condition allows us to apply Shearer’s inequality, and whileShearer’s inequality itself cannot be strengthened in its full generality to deal with non-uniform covers (inthat entropy may concentrate on the vertices covered the fewest number of times), some variant of Shearer’sinequality tailored to the situation at hand might nevertheless be an ingredient that we are presently missing.Second, a significant difference between r = 2 and r ≥ H rd is no longer vertex transitive when r ≥
3. Any analogue of the swapping trick of Zhao [7] for r -graphs with r ≥ H rd to disjoint copiesof H rd and, in the absence of symmetry, this seems difficult to accomplish . cknowledgements The first author was partially supported by NSF grant DMS-1764123, an Arnold O. Beckman ResearchAward (UIUC Campus Research Board 18132) and the Langan Scholar Fund (UIUC), the second author wassupported by NSF grant DMS-1855745, and the third author wishes to acknowledge support from NSF grantDMS-1800521.Part of this work was done while the first author was a visiting fellow commoner at Trinity College,Cambridge; we thank Trinity College for their hospitality.
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The number of independent sets in a regular graph , Combin. Probab. Comput. (2010),315–320.8. , Extremal regular graphs : independent sets and graph homomorphisms , Amer. Math. Monthly (2017), 827–843. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801
E-mail address : [email protected] Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road,Cambridge CB3 0WB, UK, and
Department of Mathematical Sciences, University of Memphis, Memphis TN 38152,USA
E-mail address : [email protected] Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
E-mail address : [email protected]@math.rutgers.edu