Counting Hamilton Cycles in Dirac Hypergraphs

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The complete bipartite graph K n,n+1 is an extremal example for the tightness of this minimum degree condition. Moreover, since adding one edge to it already creates many Hamilton cycles, this suggests that Dirac graphs might contain not only one, but many Hamilton cycles. This leads to the question of how many Hamilton cycles are contained in a Dirac graph. In a seminal paper by Sárközy et al. [20] it was proven that n-vertex Dirac graphs contain at least c n n! many distinct Hamilton cycles for some small positive constant c. As this is clearly the correct order of magnitude, one could further ask for the correct value of the constant c.
Toward an intelligent guess for the value of c, consider a binomial random graph G n, p on n vertices wherein each edge appears independently with probability p. It is easy to show (for example by Chernoff bounds) that with high probability (i.e., with probability tending to 1 as n → ∞), its minimum degree is (1 − o(1))np, and that the expected number of Hamilton cycles is (1)) n n! p n .
(1.1) (It is not so easy to show concentration though! See Janson [9].) This hints that we might take c ≈ δ(G)/n, where δ(G) denotes the the minimum degree in G. Indeed, Cuckler and Kahn [2] impressively showed that c ≈ δ(G)/n is the correct constant, thereby closing the case completely. It is natural to extend Cuckler and Kahn's result to the hypergraph setting. First, let us introduce a notion of cycle in hypergraphs. For positive integers 0 ≤ < k, we define a (k, )-cycle to be a k-uniform hypergraph (or a "k-graph" for short) whose vertices may be ordered cyclically such that its edges are segments of length k and every two consecutive edges overlap in exactly vertices. A (k, )-cycle which contains all the vertices of a given k-graph is called a Hamilton -cycle. We say that a k-graph is -Hamiltonian if it contains a Hamilton -cycle. When = k − 1 we often refer to an -cycle as a tight cycle, and we say that a k-graph is tight Hamiltonian or contains a Hamilton tight cycle, accordingly. Note that in order for a k-graph on n vertices to be -Hamiltonian, it is necessary that n be divisible by k − . In light of Dirac's theorem, we also consider the more general notion of degrees in hypergraphs. We say that the co-degree of a (k − 1)-set X in a k-graph H is the number of edges in H that contain X . There has been much work on analogues of Dirac's theorem in the hypergraph setting. Initial results were due to Katona and Kierstead [11]. Later it was shown that the necessary minimum co-degree for a k-graph H to be -Hamiltonian is δ k−1 (H ) ≈ n/2, for = k − 1 [18,19], and more generally for satisfying (k − ) | k [14]. For values of satisfying (k − ) k, it was proven in [13] that the necessary minimum co-degree for -Hamiltonicity is For more details about (many) other results regarding the minimum co-degree of a hypergraph and -Hamiltonicity, we refer the reader to the excellent surveys by Rödl and Ruciński [17] and by Kühn and Osthus [12].
In light of these results, and since we consider Hamilton -cycles for various values of , we say that H is δ-Dirac if δ k−1 (H ) ≥ δn for some constant δ > 1/2. For the sake of simplicity, we have not tried to optimize the value of δ, so we always assume that δ > 1/2 is a constant. However, with some more careful calculations and slight adjustments to the parameters involved, we can modify all our proofs so that they hold for any, say, δ ≥ 1/2 + ω (log 3 n/n) 1/5 .
A natural guess for the correct lower bound on the number of Hamilton -cycles in a δ-Dirac graph is the expected number of Hamilton -cycles in a random hypergraph with edge density δ. That is, we hope to obtain a lower bound of the form where k (n, ) is the number of Hamilton -cycles in the complete k-graph on n vertices. Ferber et al. [6] realized this hope in the case where ≤ k/2. Quite recently, Glock et al. [7] showed that a δ-Dirac k-graph contains at least (1 − o(1)) n n!c n Hamilton -cycles, for all values and for some small constant c > 0. Our contribution is that (1.2) is the correct lower bound for all values of 0 ≤ < k − 1 in any δ-Dirac k-graph.

Proof Outline
For m ∈ N and X ⊂ V (H ) we define an ( , m, X )-path-system to be an ordered collection of m many vertex-disjoint -paths that cover X , and let P( , m, X ) be the collection of all ( , m, X )-path-systems in H (see Definition 3.7). Our proof is largely based on the following three steps: (i) We remove a small subset W ⊂ V with certain properties. This set will be used to tailor path-systems into Hamilton cycles.
(ii) We show that, for an appropriate choice of m, |P( , m, V \ W )| is at least as large as the number of Hamilton -cycles we eventually want. (iii) We show that each P ∈ P( , m, V \ W ) can be tailored into a Hamilton -cycle using the vertices in W in such a way that distinct path-systems correspond to distinct cycles. Clearly, it then follows that the number of Hamilton -cycles in H is at least the size of P( , m, V \W ), as required. Section 3.4 is dedicated to steps (i) and (iii), which mainly follow from other results (mostly stated in [7]). More specifically, in Lemma 3.13 we prove that our k-graph contains such a subset W ⊂ V , and in Lemma 3.16 we show how to tailor a path-system into a Hamilton -cycle using the set W .
Our main contribution is in step (ii), where the goal is to construct "many" pathsystems, each of which covers all of the vertices in V := V \ W . As mentioned above, we use W to tailor each path-system into exactly one Hamilton -cycle. Therefore, if we let x := |W |, then we clearly cannot have more than ( (n − x)) n−x many path-systems in V , as this is at most how many Hamilton cycles one can have on n − x vertices. Since the desired lower bound on the number of Hamilton -cycles is of order ( (n)) n , we require (n − x) −x = (1 − o(1)) n . Therefore, we must choose x = o n log n . Moreover, since we need to sew m paths together using vertices from W we also must have that m = O(x).
For convenience we only count (and construct) path-systems in which all paths are of exactly the same length s := n−x m (so from the above discussion we must have s = ω(log n)). The process of constructing a path-system goes as follows. First, we choose an ordered equipartition V = V 1 ∪ . . . ∪ V s . Second, we choose an "ordered" perfect matching M 1 = (e 1 , . . . , e m ) in the k-partite k-graph induced by V 1 ∪ . . .∪ V k , and for each i ∈ [m] we let X e i be the vertices in e i that are contained in the last parts V k− +1 , . . . , V k . Next, we choose a perfect matching M 2 = ( f 1 , . . . , f m ) in the k-partite k-graph induced by V k− +1 ∪ . . . ∪ V 2k− in such a way that for each i we have f i ∩ e i = X e i , and define X f i , analogously, to be the intersection of f i with the last parts V 2(k− )+1 , . . . , V 2k− , for each i ∈ [m]. We repeat this, choosing a perfect matching M 3 = (g 1 , . . . , g m ) in the k-partite k-graph induced by V 2(k− )+1 , . . . , V 3k−2 such that g i ∩ f i = X f i and define X g i analogously. We continue this way, considering the next k parts of the partition in steps of size k − , until we cover V . Clearly, the union of all the M i 's is an ( , m, V )-path-system (the order of the paths is induced by the order on M 1 ).
Our goal is to show that this process yields many path-systems. Hence, the main building block in our counting argument will be finding many perfect matchings in kpartite k-graphs, where the intersection of each edge with the first parts is determined. Section 3.2 is dedicated to showing that this is possible when considering a δ-Dirac k-partite k-graph (where the Dirac property applies only for (k − 1)-sets with vertices in distinct parts). We prove Lemma 3.4, and as a consequence we get Corollary 3.6, showing that many perfect matchings can indeed be found in each step of this process.
Having proved our building block in Corollary 3.6, we describe in detail the process of constructing many path-systems in Sect. 3.3. We first prove Lemma 3.8, showing that for an appropriate choice of the parameter m, most of the ordered equipartitions inherit the Dirac property of our k-graph (as s-partite induced k-graphs). We then prove Lemma 3.9, showing that for each such "good" equipartition we can construct many distinct path-systems by concatenating perfect matchings from each step in process described above. We conclude the section with Corollary 3.10, where we combine both lemmata to get the "correct" number of path-systems in our k-graph.
Lastly, in Sect. 4 we tie everything together, showing that Lemma 3.13, Corollary 3.10, and Lemma 3.16 imply our result for an appropriate choice of parameters.

Concentration Inequalities
We use two probabilistic tools. The first one is the known result by Chernoff, bounding the lower and the upper tails of the Binomial distribution (see [1,10]).

Remark 3.2
The above bounds also hold when X is a hypergeometric random variable.
Our second probabilistic tool is an application of a concentration inequality by McDiarmid [16], proved originally by Maurey [15] as one of the first uses of concentration inequalities outside of probability theory. Theorem 3.3 Let S n be the group of permutations over a set of n elements, and let h : S n → R. Assume that for some constant c we have that |h(π ) − h(π )| ≤ c for any π, π ∈ S n which are obtained from one another by swapping two elements. Then for any t ≥ 0 we have

The Number of Perfect Matchings in k-Partite k-Graphs
In this section we show that in a δ-Dirac k-partite k-graph, one can find "many" perfect matchings, even when the intersection of each edge in each matching with the first parts is determined. This is our main building block for constructing many pathsystems in the proof of the main result. We show this as a corollary of a more general statement, Lemma 3.4, which is a consequence of the concentration inequality given by Theorem 3.3. The focus of this section is proving Lemma 3.4, which allows us to reduce the problem to the bipartite case in graphs. This is a version of an idea from [4]. Then, by using a known result by Cuckler and Kahn [2], we deduce Corollary 3.6.
We start by introducing some further definitions and notation. Let H be a k-partite kgraph with parts V 1 , . . . , V k , all of size m. Let π = (π 1 , . . . , π k−1 ), where π i : [m] → V i is a permutation on the vertices in V i for each i ∈ [k − 1]. Let M π be the collection of (k − 1)-sets that intersect all V 1 , . . . , V k−1 , induced by π . More precisely, let Define the auxiliary graph B π (H ) to be the bipartite graph with parts M π and V k , and such that xv is an edge for If π k : [m] → V k is a permutation of the vertices in V k , we say that the tuple of k permutations (π 1 , . . . , π k ) induces a perfect matching in H , if the set of edges When considering an s-partite k-graph H , for some s ≥ k, it is simpler to use the following variant of the notion of minimum co-degree. Assume that V 1 , . . . , V s are the parts of H . For i ∈ [s] define where the union goes over all is the minimum number of edges incident to a (k − 1)-set that intersects exactly k − 1 parts.
We can now state the main lemma of this section.

Lemma 3.4
For every ε > 0 there exists m 0 ∈ N such that the following holds for be a permutation on the vertices of V i , such that π 1 , . . . , π k−2 are fixed and π k−1 is chosen uniformly at random. Denote π := (π 1 , . . . , π k−1 ). Then with high probability the bipartite graph B π (H ) has minimum degree at least (δ − ε)m.
Proof Recall that the graph B π := B π (H ) has parts M π and V k , and note that for Thus we have Now note that swapping any two elements in π k−1 can change d B π (v) by at most 2. Thus, by Theorem 3.3 we get that Hence, after taking a union bound over all vertices in V k , we see that the minimum degree in B π (H ) is at least (δ − ε)m with high probability.
In [2] the authors provide a lower bound on the number of perfect matchings in Dirac graphs, given naturally by the lower bound on the number of Hamilton cycles in those graphs.

Theorem 3.5 (Theorems 1.5 and 3.1 in [2]) Let G be a bipartite Dirac graph on parts of size n, and with minimum degree d ≥ n/2. Then G contains at least
(1 − o(1)) n · n! · d n n many perfect matchings.
We combine Lemma 3.4 and Theorem 3.5 to get the following corollary.  (1)) m (m!) k−r δ m many tuples π = (π 1 , . . . , π k ) which induce a perfect matching in H (where one perfect matching may be induced by several tuples of permutations).

Constructing Many ( , m, V )-Path-Systems
This section is the heart of the argument. We show that we can cover a subset containing most of the vertices in H by the "correct" number of path-systems, that is, the number of Hamilton -cycles we aim to find in H . We start with the precise notion of a pathsystem. Similarly to an -cycle, an -path is a k-graph whose vertices may be ordered so that its edges are segments of length k and consecutive edges overlap in exactly vertices. Note that two different ( , m, X )-path-systems may consist of the same family ofpaths but with different orderings. We distinguish between two such families, since they will eventually form two different Hamilton -cycles when we sew the paths to one another. For a partition of the vertices of a k-graph H into s parts, we denote by H [ ] the s-partite k-graph spanned by edges going between parts of . We use the following two lemmata to show that a subset containing most of the vertices in a δ-Dirac k-graph can be covered by many path-systems. (3.1) Proof Let 0 < ε < δ/2 and let n be sufficiently large. Since V contains all but o(n) many vertices in H , and since δ k−1 (H ) ≥ δn, we get Since d(X , V i ) has hypergeometric distribution for each i, by Lemma 3.1 we get that Taking a union bound over all i ∈ [n /m] and X ∈ V k−1 we get and the statement follows.
In other words, Lemma 3.8 shows that almost all partitions of H [V ] into n /m parts inherit the relative minimum co-degree from H as induced (n /m)-partite k-graphs. In particular, if H is δ-Dirac, then most of these partitions inherit this property (perhaps with a slightly smaller value of δ). Now we show that we may cover those induced k-graphs given by "good" partitions with many path-systems.

1). Then the n m -partite k-graph H [ ] can be covered by at least
Proof For the first step we consider the k-partite subhypergraph of H induced by the first k parts of , that is H 1 := H [V 1 , . . . , V k ]. By Corollary 3.6 with r = 0 we get that there are at least many π 1 = (π 1 1 , . . . , π 1 k ) which give a perfect matching in H 1 . Fix one such π 1 = (π 1 1 , . . . , π 1 k ), and let M 1 be the ordered perfect matching induced by π 1 , ordered according to π 1 1 . That is, Moreover, for each j ∈ [m], let X 1 j be the intersection of e 1 j with parts V k− +1 , . . . , V k , that is, For the second step, consider the k consecutive parts in beginning with the last parts of H 1 . More precisely, we look at the k-partite induced k-graph H 2 := H [V k− +1 , . . . , V 2k− ], and we find there a perfect matching M 2 = (e 2 1 , . . . , e 2 m ) (ordered according to π 1 k− +1 ) that extends M 1 in the sense that e 2 j ∩ e 1 j = X 1 j for every j ∈ [m]. We do this as follows. We let (π 2 1 , . . . , π 2 ) = (π 1 k− +1 , . . . , π 1 k ) be our fixed permutations on V k− +1 , . . . , V k , respectively. Then, by Corollary 3.6 with r = , there are at least many tuples (π 2 +1 , . . . , π 2 k ) for which π 2 = (π 2 1 , . . . , π 2 k ) induces a perfect matching in H 2 . Note further that two distinct such tuples induce two distinct perfect matchings in H 2 , since π 2 1 is fixed. Hence, we get that the number of perfect matchings in H 2 which agree with M 1 on the first parts is at least what is given in (3.3). Let M 2 = (e 2 1 , . . . , e 2 m ) be one such perfect matching, and for every j ∈ [m], let X 2 j to be the intersection of e 2 j with the last parts of H 2 , that is Note that indeed we have e 2 j ∩ e 1 j = X 1 j for every j ∈ [m]. We then repeat the above procedure, where in each step we consider the next k parts, overlapping with the last parts from the preceding step. We do this n /m−k k− times, until we cover all parts in .
Combining the counting in Lemmas 3.8 and in 3.9, we get the following corollary, which gives us the required number of ( , m, V )-path-systems covering H [V ]. However, for some values of , the same path-system can be obtained from many different orderings of certain vertices within the paths. We denote by c k ( ) the number of ways to reorder the first k − vertices in any edge such that the path-system is not changed, so we get (c k ( )) n k− many such reorderings. Considering this double counting, and recalling that n = n − o n log n , we get that in total H [V ] can be covered by at least In order to show that this is precisely the expression in (3.4), it is sufficient to prove the following claim.
Proof In the complete k-graph on n vertices, any cyclical ordering of the vertices yields a Hamilton -cycle (where n, k, and satisfy the appropriate divisibility conditions). However, several different vertex-orderings may give rise to the same cycle. For any fixed ordering, we may of course cyclically permute the edges in n k− ways or reverse their ordering and leave the cycle unchanged. Additionally, by the definition of c k ( ), we may reorder the first k − vertices within each edge in c k ( ) ways per edge and obtain the same cycle. So we have To complete the proof, we also describe the quantity c k ( ) explicitly. Let 0 ≤ r < k− be the remainder when dividing k by k− . That is, r satisfies k = (k− ) k k− +r . Then we have (3.5) For example, for < k/2 we have r = , which gives c k ( ) = !(k − 2 )!. Note that, in particular, we have k (n, ) = n (n) .
All in all, we get that there are at least

Turning ( , m, V )-Path-Systems into Hamilton -Cycles
In this section we show how to form a Hamilton -cycle from an ( , m, V )-pathsystem. We do this in three steps. First, we put aside a special subset of vertices. Second, we cover the remaining vertices with a path-system. Finally, we use our special subset to tailor together the paths in the path-system so that they form a Hamilton -cycle.

Finding a (ı − o(1), , m, t)-Connecting-Set in a Dirac Hypergraph
Let us first make explicit the properties of the special set to put aside. We will actually put aside a family of subsets-one subset for each path in the path-system. F be a k-graph, η ∈ (0, 1), and , m, t be integers such that 1 ≤ ≤ k − 1 and 1 ≤ m ≤ t. We say that an ordered collection

Definition 3.12 Let
In other words, an (η, , m, t)-connecting-system consists of t vertices (property (ii)), broken into m parts (property (i)) of roughly equal size (property (iv)), where each (k − 1)-set in F completes to an edge with an η-fraction of each part (property (v)). We will see later that property (iii) ensures that some divisibility conditions are met when we create our -cycles.
We now show that we can find such a set W in H with η ≈ δ. The proof is similar to that of Lemma 3.8. For every X ∈ V k−1 and for every i ∈ [m], the co-degree d(X , W i ) is a hypergeometrically distributed random variable with Fix X ∈ V k−1 , and i ∈ [m]. By Lemma 3.1 with 0 < ε < δ and by the above, we have Hence, property (v) is satisfied, so with high probability W is a (δ − ε, , m, t)connecting-system in H . Using a (ı − o(1), , m, t)

-Connecting-System
It is left to show how to use a (δ − o(1), , m, t)-connecting-system in H , for appropriate parameters m, t, to form a Hamilton -cycle from an ( , m, V )-path-system that covers the rest of the vertices in H . For this we need Lemma 3.15, which follows from Lemma 3.6 in [7], and the notion of -Hamiltonian connectedness. Given a Hamilton -path P = (x 1 , . . . , x n ), we say that (x 1 , . . . , x ) and (x n− +1 , . . . , x n ) are the -end-tuples of P.
Definition 3.14 A k-graph F is -Hamiltonian connected if, for every two disjoint ordered subsets of vertices The authors of [7] show that all sufficiently dense (in the sense of co-degree) kgraphs on sufficiently many vertices are (k − 1)-Hamiltonian connected (provided that divisibility conditions are met). Their proof implies the next lemma which is slightly more general.

Lemma 3.15
For every ε > 0 there exists n 0 such that every k-graph F on n ≥ n 0 vertices, where n ≡ k(modk − ), and with minimum co-degree at least (1/2 + ε)n is -Hamiltonian connected.
Now we show that when we put aside a connecting-system and cover the remaining vertices with a path-system, we may connect the paths into Hamilton -cycles in such a way that distinct path-systems yield distinct cycles. Proof Let W = W 1 ∪ · · · ∪ W m be a partition of W such that W = (W 1 , . . . , W m ) is a (δ − ε, , m, t)-connecting-system in H , for some 0 < ε < δ/2 − 1/4. We use these parts to tailor the -paths in P to one another. For every i ∈ [m] let − → X i and − → Y i be the -end-tuples of the path P i , and let X i , Y i be their unordered sets of vertices, respectively. By Definition 3.12 (v), for each i ∈ [m], the induced subgraph H i := H [W i ∪ Y i ∪ X i+1 ] (where we replace i + 1 by 1 when i = m) has minimum co-degree at least (δ −2ε)|H i | > 1 2 |H i |. By items (ii) and (iv) of Definition 3.12 we get that |H i | = t m = ω(1) for each i ∈ [m]. Moreover, since |W i | ≡ −k(modk − ) and |X i+1 | = |Y i | = , we get that |H i | ≡ k(modk − ) for every i ∈ [m]. Hence, by Lemma 3.15 it is -Hamiltonian connected. Let Q i be a Hamilton -path in H i with − → Y i and − → X i+1 as -end-tuples. Then we get that C := (P 1 , Q 1 , . . . , P m , Q m ) is a Hamilton -cycle in H .
Note that in C, any two paths P i , P i+1 are separated by the vertices of W i , where W is fixed in the process. Hence, given W, any two distinct ( , m, V )-path-systems form two distinct Hamilton -cycles.

Proof of Theorem 1.1
We can now put all of the ingredients together and quickly derive the proof of our main theorem.

Concluding Remarks and Open Problems
We highlight the natural barrier in our approach for extending our main result to tight Hamilton cycles. The main obstacle is extending Corollary 3.6, a critical component of our proof, to the case where r = k − 1. More precisely, in the setting of Corollary 3.6, we cannot use Theorem 3.3 to find many perfect matchings in B π (H ) for fixed π = (π 1 , . . . , π k−1 ) because we no longer have a "free" part where we can permute vertices at random. In order to resolve the tight case, we believe that a different approach is necessary. However, it seems reasonable to believe that the number of tight Hamilton cycles one can find in a δ-Dirac k-graph is consistent with our result for all other values of . Hence we state here the following conjecture, which is a slight generalization of Conjecture 7.1 in [7].