Hamiltonicity in randomly perturbed hypergraphs

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Abstract

For integers k3 and 1k1, we prove that for any α>0, there exist ϵ>0 and C>0 such that for sufficiently large n(k)N, the union of a k-uniform hypergraph with minimum vertex degree αnk1 and a binomial random k-uniform hypergraph G(k)(n,p) with pn(k)ϵ for 2 and pCn(k1) for =1 on the same vertex set contains a Hamiltonian -cycle with high probability. Our result is best possible up to the values of ϵ and C and answers a question of Krivelevich, Kwan and Sudakov.

Introduction

The study of Hamiltonicity (the existence of a spanning cycle) has been a central and fruitful area in graph theory. In particular, a celebrated result of Karp [19] states that the decision problem for Hamiltonicity in graphs is NP-complete. So it is desirable to study sufficient conditions that guarantees Hamiltonicity. Among a large variety of such results, probably the most well-known is a theorem of Dirac from 1952 [11]: every n-vertex graph (n3) with minimum degree at least n/2 is Hamiltonian.

Another well-studied object in graph theory is the random graph G(n,p), which contains n vertices and each pair of vertices forms an edge with probability p independently from other pairs. Pósa [27] and Korshunov [21] independently determined the threshold for Hamiltonicity in G(n,p), which is around logn/n. This implies that almost all dense graphs are Hamiltonian. Furthermore, Bohman, Frieze and Martin [6] showed that for every α>0 there is c=c(α) such that every n-vertex graph G with minimum degree αn becomes Hamiltonian a.a.s. after adding cn random edges (we say that an event happens asymptotically almost surely, or a.a.s., if it happens with probability 1o(1)). This result is tight up to the value of c by considering a complete bipartite graph Kαn,(1α)n. A comparison can be drawn to the notion of smoothed analysis of algorithms introduced by Spielman and Teng [34], which involves studying the performance of algorithms on randomly perturbed inputs.

It is natural to study the Hamiltonicity of uniform hypergraphs. Given k2, a k-uniform hypergraph (in short, a k-graph) H=(V,E) consists of a vertex set V and an edge set E(Vk), where every edge is a k-element subset of V. Given a k-graph H with a set S of d vertices (where 1dk1) we define NH(S) to be the collection of (kd)-sets T such that STE(H), and let degH(S):=|NH(S)|. The minimum d-degree δd(H) of H is the minimum of degH(S) over all d-vertex sets S in H.

In the last two decades, there has been a growing interest of extending Dirac's theorem to hypergraphs. Despite other notion of cycles in hypergraphs (e.g., Berge cycles), the following definition of cycles has become more popular recently (see surveys [29], [35]). For integers 1k1 and m3, a k-graph F with m(k) vertices and m edges is called an ℓ-cycle if its vertices can be ordered cyclically such that each of its edges consists of k consecutive vertices and every two consecutive edges (in the natural order of the edges) share exactly vertices. A k-graph is called ℓ-Hamiltonian if it contains an -cycle as a spanning subgraph. Extending Dirac's theorem, the minimum d-degree conditions that force -Hamiltonicity (for 1d,k1) have been intensively studied [2], [3], [9], [10], [14], [17], [15], [16], [20], [24], [25], [30], [31], [32], [33]. For example, the minimum 1-degree threshold for 2-Hamiltonicity in 3-graphs was determined asymptotically [28].

Let G(k)(n,p) denote the binomial random k-graph on n vertices, where each k-set forms an edge independently with probability p. The thresholds for -Hamiltonicity have been studied by Dudek and Frieze [13], [12], who proved that the asymptotic threshold is 1/nk for 2 and logn/nk1 for =1 (they also gave a sharp threshold for k4 and =k1).

It is also natural to consider -Hamiltonicity in randomly perturbed k-graphs. In fact, Krivelevich, Kwan and Sudakov [22] extended the result of Bohman–Frieze–Martin [6] to hypergraphs.

Theorem 1.1

[22] Let kN, and let H be a k-graph on n(k1)N vertices with δk1(H)αn. There exists a function ck=ck(α) such that for p=ck/nk1, HG(k)(n,p) a.a.s. is 1-Hamiltonian.

Theorem 1.1 is tight up to the value of ck (see the paragraph after Theorem 1.2). Similar results for the powers of Hamiltonian (k1)-cycles were obtained by Bennett, Dudek and Frieze [5], and recently by Bedenknecht, Han, Kohayakawa and Mota [4]. In addition, Böttcher, Montgomery, Parczyk and Person [8] proved embedding results for bounded degree subgraphs in randomly perturbed graphs. Other results in randomly perturbed graphs can be found in [1], [23], [7].

Krivelevich, Kwan and Sudakov [22] asked whether Theorem 1.1 can be extended to -Hamiltonicity under minimum d-degree conditions for 1d,k1. McDowell and Mycroft [26] found such results for dmax{,k} and reiterated the question for arbitrary d and . In this paper we solve this problem completely. Since the minimum 1-degree condition is the weakest among d-degree conditions for all d1, we only state and prove our result with respect to the minimum 1-degree.

Theorem 1.2

For integers k3, 1k1 and α>0 there exist ϵ>0 and an integer C>0 such that the following holds for sufficiently large n(k)N. Suppose H is a k-graph on n vertices with δ1(H)αnk1 andp=p(n){n(k)ϵif2,Cn(k1)if=1. Then HG(k)(n,p) a.a.s. is ℓ-Hamiltonian.

Theorem 1.2 is sharp up to the constants ϵ and C. Indeed, given k and , let α>0 be sufficiently small and n(k)N be sufficiently large. Consider a partition AB of a vertex set V such that |A|=αn and |B|=(1α)n. Let H0 be the k-graph with all k-tuples that intersect both A and B as edges. It is easy to see that δ1(H0)=αn(nαn1k2). Suppose H0G(k)(n,p) a.a.s. contains a Hamiltonian -cycle C. Since |A|=αn, C contains at least 1/α1 consecutive vertices in B. Let a=(1/α1)/(k). Since B is an independent set in H0, this implies that G(k)(n,p) a.a.s. contains an ℓ-path on a edges (a k-graph with vertices v1,v2,,va(k)+ and edges {vi(k)+1,,vi(k)+k} for i=0,,a1). When p<(1/2)1/an(k)/a, we have n+a(k)pa<1/2. By Markov's inequality, with probability at least 1/2, G(k)(n,p) contains no -path on a edges. When =1, if H0G(k)(n,p) is a.a.s. ℓ-Hamiltonian, then G(k)(n,p) a.a.s. contains n/(k1)2|A|>n/k edges (because a 1-Hamiltonian cycle contains n/(k1) edges and each vertex is contained in at most 2 of them). When p<n(k1)/(2k), we have nkp<n/(2k). By Markov's inequality, with probability at least 1/2, G(k)(n,p) contains fewer than n/k edges.

The proof of Theorem 1.2 follows the absorbing method introduced by Rödl, Ruciński, and Szemerédi in [31]. Let us define absorbers for our problem. Given an -path P, we call the first and last vertices two ℓ-ends of P. Let H be a k-graph and S be a set of k vertices in V(H). We call an -path P an S-absorber if V(P)S= and V(P)S spans an -path with the same ℓ-ends as P.

Below is a typical procedure for finding a Hamilton -cycle in H by the absorbing method.

  • (1)

    We show that every (k)-subset of V(H) has many absorbers (of the same fixed length). This enables us to obtain a path Pabs of linear length such that every (k)-set has many absorbers on Pabs.

  • (2)

    We cover most vertices of VV(Pabs) by short paths and then connect them together with Pabs into a cycle C.

  • (3)

    The vertices not covered by C are arbitrarily partitioned into (k)-sets and absorbed by Pabs greedily.

The proof thus has three main components:
  • an absorbing lemma, which provides a family A of vertex-disjoint short paths such that every (k)-set has many absorbers in A;

  • a path cover lemma, which allows us to cover most vertices of V(H) by vertex-disjoint paths; and

  • a connecting lemma, which allows us to connect A into a single path Pabs and connect the paths from the path cover lemma together.

Let G(k)(n,p)H be the underlying k-graph on the same vertex set V. Using Janson's inequality, one can derive the path cover lemma by using the edges of G(k)(n,p). If we have δk(H)α(n), then every (k)-set of V has many neighbors and it is not difficult to prove the absorbing lemma. If we have δ(H)α(nk), then every -set of V has many neighbors and it is easy to prove the connecting lemma. However, our Theorem 1.2 only assumes δ1(H)αnk1. In order to prove Theorem 1.2, we “shave” H by removing all the edges of H that contain an -set of low degree. This results in a k-graph H in which every -subset of V either has a high degree or a zero degree. Our connecting lemma only connects two -sets with high degree. To overcome the difficulty in absorbing, an earlier version of this paper used the hypergraph regularity method. Following the suggestion of a referee, we now give a simpler absorbing lemma without the regularity method. Note that the shaving process creates a small number of vertices that cannot be absorbed and we will cover these vertices by the path cover lemma.

The rest of the paper is organized as follows. We state and prove our lemmas in Sections 2 and 3 and prove Theorem 1.2 in Section 4.

Notation. Given positive integers nb, let [n]:={1,2,,n} and (n)b:=n(n1)(nb+1)=n!/(nb)!. Given a k-graph H, we use vH and eH to denote the order and size of H, respectively. For two (hyper)graphs G and H, let GH (or GH) denote the (hyper)graph with vertex set V(G)V(H) (or V(G)V(H)) and edge set E(G)E(H) (or E(G)E(H)). Given a set X, (Xk) denotes the family of all k-subsets of X. A k-graph (V,E) is complete if E=(Vk). Given 1k, the ℓ-shadow of a k-graph H, denoted by H, is the collection of all -subsets SV(H) that are contained in some edges of H.

In this paper, unless stated otherwise, we assume that the vertex sets of paths and related hypergraphs are ordered. When A and B are ordered sets, let AB denote their concatenation. Given positive integers k,,a such that <k, let Pa denote a k-uniform ℓ-path of length a, that is, a k-graph on vertices v1,v2,,va(k)+ with edges {vi(k)+1,,vi(k)+k} for i=0,,a1. In general, given a k-graph F on {x1,,xs} and a k-graph H, we say that an ordered subset (v1,,vs) of V(H) spans a (labeled) copy of F if {vi1,,vik}E(H) whenever {xi1,,xik}E(F). Given integers a1 and x0, let Pa,x denote a k-graph on a(k)++2x vertices with an order such that the first and last x vertices are isolated and the middle a(k)+ vertices span a copy of Pa.

Throughout the rest of the paper, we write αβγ to mean that we can choose the positive constants α,β,γ from right to left. More precisely, there are increasing functions f and g such that, given γ, whenever βf(γ) and αg(β), the subsequent statement holds. Hierarchies of other lengths are defined similarly.

Throughout the paper we omit floor and ceiling functions when they are not crucial.

Section snippets

Subgraphs in random hypergraphs

In this section we introduce some results related to binomial random k-graphs (similar ones can be found in [4]). Our main tools are Janson's inequality (see, e.g., [18, Theorem 2.14]) and Chebyshev's inequality.

We first recall Janson's inequality. Let Γ be a finite set and let Γp be a random subset of Γ such that each element of Γ is included independently with probability p. Let S be a family of non-empty subsets of Γ and for each SS, let IS be the indicator random variable for the event SΓp

Lemmas

In this section we prove all the lemmas that are needed for the proof of Theorem 1.2.

Since we assume δ1(H)αnk1, unless =1, the k-graph H may contain some -sets S whose degree is too low to be used for connection. To overcome this, we simply delete all edges that contain S. The following lemma reflects this “shaving” process.

Lemma 3.1

Let 0<ηα,1/k. Let H be an n-vertex k-graph with δ1(H)αnk1. Then there exists a spanning subgraph H of H, satisfying the following properties.

  • (1)

    e(H)αnk/(2k).

  • (2)

    degH(v)

Proof of Theorem 1.2

In this section we prove Theorem 1.2. We essentially follow the procedure mentioned in Section 1.3 but need additional work. We first apply Lemma 3.1 and obtain a spanning subgraph H of H. Let V be the set of vertices of H with high degree. Following the procedure outlined in Section 1.3, we obtain an absorbing path Pabs, a set C1 of connectors and a set P of paths that cover almost all the vertices. A natural attempt is to use the connectors in C1 to connect the paths in P and Pabs to

Acknowledgement

We would like to thank Wiebke Bedenknecht, Yoshiharu Kohayakawa and Guilherme Mota for discussions at an early stage of this project. We are also grateful to two anonymous referees for many helpful comments. In particular, we are in debt to a referee who showed us how to obtain an absorbing lemma without using the regularity method. This and other comments helped to simplify our proof and greatly improved the presentation of the paper.

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    JH was partially supported by FAPESP (Proc. 2014/18641-5) and Simons Foundation #630884. YZ is partially supported by NSF grant DMS 1700622.

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