Hamiltonicity in randomly perturbed hypergraphs☆
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 () with minimum degree at least is Hamiltonian.
Another well-studied object in graph theory is the random graph , 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 , which is around . This implies that almost all dense graphs are Hamiltonian. Furthermore, Bohman, Frieze and Martin [6] showed that for every there is 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 ). This result is tight up to the value of c by considering a complete bipartite graph . 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 , a k-uniform hypergraph (in short, a k-graph) consists of a vertex set V and an edge set , where every edge is a k-element subset of V. Given a k-graph H with a set S of d vertices (where ) we define to be the collection of -sets T such that , and let . The minimum d-degree of H is the minimum of 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 and , a k-graph F with 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 ℓ vertices. A k-graph is called ℓ-Hamiltonian if it contains an ℓ-cycle as a spanning subgraph. Extending Dirac's theorem, the minimum d-degree conditions that force ℓ-Hamiltonicity (for ) 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 denote the binomial random k-graph on n vertices, where each k-set forms an edge independently with probability p. The thresholds for ℓ-Hamiltonicity have been studied by Dudek and Frieze [13], [12], who proved that the asymptotic threshold is for and for (they also gave a sharp threshold for and ).
It is also natural to consider ℓ-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 , and let H be a k-graph on vertices with . There exists a function such that for , a.a.s. is 1-Hamiltonian.
Theorem 1.1 is tight up to the value of (see the paragraph after Theorem 1.2). Similar results for the powers of Hamiltonian -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 ℓ-Hamiltonicity under minimum d-degree conditions for . McDowell and Mycroft [26] found such results for and reiterated the question for arbitrary d and ℓ. In this paper we solve this problem completely. Since the minimum 1-degree condition is the weakest among d-degree conditions for all , we only state and prove our result with respect to the minimum 1-degree.
Theorem 1.2 For integers , and there exist and an integer such that the following holds for sufficiently large . Suppose H is a k-graph on n vertices with and Then a.a.s. is ℓ-Hamiltonian.
Theorem 1.2 is sharp up to the constants ϵ and C. Indeed, given k and ℓ, let be sufficiently small and be sufficiently large. Consider a partition of a vertex set V such that and . Let be the k-graph with all k-tuples that intersect both A and B as edges. It is easy to see that . Suppose a.a.s. contains a Hamiltonian ℓ-cycle C. Since , C contains at least consecutive vertices in B. Let . Since B is an independent set in , this implies that a.a.s. contains an ℓ-path on a edges (a k-graph with vertices and edges for ). When , we have . By Markov's inequality, with probability at least 1/2, contains no ℓ-path on a edges. When , if is a.a.s. ℓ-Hamiltonian, then a.a.s. contains edges (because a 1-Hamiltonian cycle contains edges and each vertex is contained in at most 2 of them). When , we have . By Markov's inequality, with probability at least 1/2, contains fewer than 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 ℓ-path P, we call the first and last ℓ vertices two ℓ-ends of P. Let H be a k-graph and S be a set of vertices in . We call an ℓ-path P an S-absorber if and spans an ℓ-path with the same ℓ-ends as P.
Below is a typical procedure for finding a Hamilton ℓ-cycle in H by the absorbing method.
- (1)
We show that every -subset of has many absorbers (of the same fixed length). This enables us to obtain a path of linear length such that every -set has many absorbers on .
- (2)
We cover most vertices of by short paths and then connect them together with into a cycle C.
- (3)
The vertices not covered by C are arbitrarily partitioned into -sets and absorbed by greedily.
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an absorbing lemma, which provides a family of vertex-disjoint short paths such that every -set has many absorbers in ;
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a path cover lemma, which allows us to cover most vertices of by vertex-disjoint paths; and
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a connecting lemma, which allows us to connect into a single path and connect the paths from the path cover lemma together.
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 , let and . Given a k-graph H, we use and to denote the order and size of H, respectively. For two (hyper)graphs G and H, let (or ) denote the (hyper)graph with vertex set (or ) and edge set (or ). Given a set X, denotes the family of all k-subsets of X. A k-graph is complete if . Given , the ℓ-shadow of a k-graph H, denoted by , is the collection of all ℓ-subsets 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 such that , let denote a k-uniform ℓ-path of length a, that is, a k-graph on vertices with edges for . In general, given a k-graph F on and a k-graph H, we say that an ordered subset of spans a (labeled) copy of F if whenever . Given integers and , let denote a k-graph on vertices with an order such that the first and last x vertices are isolated and the middle vertices span a copy of .
Throughout the rest of the paper, we write to mean that we can choose the positive constants from right to left. More precisely, there are increasing functions f and g such that, given γ, whenever and , 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 be a random subset of Γ such that each element of Γ is included independently with probability p. Let be a family of non-empty subsets of Γ and for each , let be the indicator random variable for the event
Lemmas
In this section we prove all the lemmas that are needed for the proof of Theorem 1.2.
Since we assume , unless , the k-graph H may contain some ℓ-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 . Let H be an n-vertex k-graph with . Then there exists a spanning subgraph of H, satisfying the following properties. .
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 of H. Let be the set of vertices of with high degree. Following the procedure outlined in Section 1.3, we obtain an absorbing path , a set of connectors and a set of paths that cover almost all the vertices. A natural attempt is to use the connectors in to connect the paths in and 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.