DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN UNIFORM HYPERGRAPHS

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1 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN UNIFORM HYPERGRAPHS Abstract. A classic result of G. A. Dirac in graph theory asserts that every n-vertex graph n 3 with minimum degree at least n/2 contains a spanning so-called Hamilton cycle. G. Y. Katona and H. A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive modulo n vertices in the ordering form an edge. Moreover, the minimum degree is the minimum k -degree, i.e. the minimum number of edges containing a fixed set of k vertices. V. Rödl, A. Ruciński, and E. Szemerédi verified approximately the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum k -degree /2 + on contains such a tight Hamilton cycle. We study the similar question for Hamilton l-cycles. A Hamilton l-cycle in an n-vertex, k-uniform hypergraph l < k is an ordering of the vertices and an ordered subset of the edges such that each such edge corresponds to k consecutive modulo n vertices and two consecutive edges intersect in precisely l vertices. We prove sufficient minimum k -degree conditions for Hamilton l- cycles if l < k/2. In particular, we show that for every l < k/2 every n-vertex, k-uniform hypergraph with minimum k -degree /2k l +on contains such a loose Hamilton l-cycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus for l =, who verified it when k = 3. Our proof is based on the so-called weak regularity lemma for hypergraphs and follows the approach of V. Rödl, A. Ruciński, and E. Szemerédi.. Introduction We consider k-uniform hypergraphs H, that are pairs H = V, E with vertex sets V = V H and edge sets E = EH V k, where V k denotes the family of all k-element subsets of the set V. We often identify a hypergraph H with its edge set, i.e. H V k. Given a k-uniform hypergraph H = V, E and a set S V s let degs denote the number of edges of H containing the set S and let δ s H be the minimum s-degree of H, i.e. the minimum of degs over all s-element sets S V. A k-uniform hypergraph is called an l-cycle if there is a cyclic ordering of the vertices such that every edge consists of k consecutive vertices, every vertex is contained in an edge and two consecutive edges where the ordering of the edges is inherited from the ordering of the vertices intersect in exactly l-vertices. Naturally, we say that a k-uniform, n-vertex hypergraph H contains a Hamilton l-cycle if there is a subhypergraph of H which forms an l-cycle and which covers all vertices of H. Note that it is necessary that k l divides n which we indicate by n k ln. The first author is supported by DFG within the research training group Methods for Discrete Structures.

2 2 We study sufficient conditions on δ k H for the existence of Hamilton l-cycles in k-uniform hypergraphs H. This research was initiated by G. Y. Katona and H. A. Kierstead [4]. These authors considered the case l = k and such l- cycles are sometimes called tight cycles. Katona and Kierstead proved that the condition δ k H 5 2k V H k + 4 2k implies the existence of a tight Hamilton path in a k-uniform hypergraph H. The same authors suggested that, in fact, δ k H n k + 2/2 should suffice and they gave a matching lower bound construction. Recently, Rödl, Ruciński, and Szemerédi [2, 4] answered the question of Katona and Kierstead approximately and showed the following. Theorem Rödl, Ruciński, & Szemerédi. For every k 3 and γ > 0 there exists an n 0 such that every k-uniform hypergraph H = V, E on V = n n 0 vertices with δ k H /2 + γn contains a Hamilton k -cycle. We focus on loose cycles, that is l-cycles for l < k/2. In this setting an edge of an l-cycle only intersects its preceding and its following edge in the cycle. Also note that if n k ln, i.e. n is a multiple of k l, then a Hamilton k - cycle contains a Hamilton l-cycle. Consequently, the minimum degree condition for l-cycles is bounded by the degree condition for k -cycles. The first result considering loose Hamilton -cycles for 3-uniform hypergraphs is due to Kühn and Osthus [9]. Theorem 2 Kühn & Osthus. For every γ > 0 there exists an n 0 such that every 3-uniform hypergraph H = V, E on V = n n 0 vertices with n even and δ 2 H /4 + γn contains a Hamilton -cycle. Kühn and Osthus also showed that this result is best possible up to the error term γn see Fact 4 below and conjectured that δ k H 2k +on should force Hamilton -cycles in k-uniform hypergraphs. We verify this conjecture and prove, more generally, the analogous result for l-cycles with l < k/2. Theorem 3 Main result. For all integers k 3 and l < k/2 and every γ > 0 there exists an n 0 such that every k-uniform hypergraph H = V, E on V = n n 0 vertices with n k ln and δ k H 2k l + γn contains a Hamilton l-cycle. For the case l = this bound was proved independently by Keevash, Kühn, Mycroft and Osthus [6]. However, their approach uses the Blow-up lemma for hypergraphs [5] and is subtantially different from ours which is based on the weak hypergraph regularity lemma, Theorem 4, and the absorption technique of Rödl, Ruciński, and Szemerédi introduced in [2]. The Theorem 3 is approximately best possible as the following straightforward extension of a construction from [9] shows. Fact 4. For every l < k/2 and n 2k ln there exists a k-uniform n hypergraph H = V, E on V = n vertices with δ k H 2k l, which contains no Hamilton l-cycle. Proof. Consider the following k-uniform hypergraph H = V, E. Let A B = V be a partition of V with A = n 2k l and let E be the set of all k-tuples from V with at least one vertex in A. Clearly, δ k H = A = n 2k l. Now consider an arbitrary cycle in H. Since l < k/2 every vertex, in particular every vertex from

3 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS 3 A, is contained in at most 2 edges of this cycle. Moreover, every edge of the cycle must intersect A. Consequently, the cycle contains at most 2 A < n/k l edges and, hence, cannot be a Hamilton cycle. We note that the construction from Fact 4 also works in the case l = k/2 for even k. However, for that case a better construction is known. More generally, if k l divides k and n kn, then a Hamilton l-cycle contains a perfect matching. Lower and upper bounds for sufficient conditions on the minimum k -degree for perfect matchings were studied in [0,, 5, 3]. In particular, a simple construction shows that δ k H n/2 k is necessary for perfect matchings and, hence, the same condition is required for Hamilton l-cycles, if k l divides k. On the other hand, Theorem shows that this condition is also approximately sufficient, thus, leaving only the case when k is not a multiple of k l and l > k/2 open. For this case, a similar construction as given in Fact 4 combined with Theorem shows that for l < k arbitrary the sufficient minimum k -degree condition lies between n k/k l k l and 2 + o n. Very recently it was shown by Kühn, Mycroft, and Osthus [8] that, indeed, if k is not a multiple of k l, then the lower bound is approximately sufficient. 2. Proof of the main result The proof of Theorem 3 follows the approach of Rödl, Ruciński, and Szemerédi from [2] and will be given in Section 2.3. This approach is based on three auxiliary lemmas, which we introduce in Section 2.2. We start with an outline of the proof. 2.. Outline of the proof. We will build the Hamilton l-cycles by connecting l-paths. An l-path with distinguished ends is defined similarly to l-cycles. Formally, a k-uniform hypergraph P is an l-path if there is an ordering v 0,..., v t of its vertices such that every edge consists of k consecutive vertices and two consecutive edges interesect in exactly l vertices. The ordered l-sets F beg = v 0,..., v l and F end = v t l,..., v t are called the ends of P. Note that this require that t l is a multiple of k l. Furthermore, for loose paths i.e. l < k/2 the ordering of the ends of an l-path do not matter and we may refer to F beg and F end as sets. The first lemma, the Absorbing Lemma Lemma 5, asserts that for l < k/2 every n-vertex, k-uniform hypergraph H = V, E with δ k H εn contains a special, so-called absorbing, l-path P, which has the following property: For every set U V \ V P with U k ln and U αn for some appropriate 0 < α ε there exists an l-path Q with the same ends as P, which covers precisely the vertices V P U. The Absorbing Lemma reduces the problem of finding a Hamilton l-cycle to the simpler problem of finding an almost spanning l-cycle, which contains the absorbing path P and covers at least αn of the vertices. We approach this simpler problem as follows. Let H be the induced subhypergraph H, which we obtain after removing the vertices of the absorbing path P guaranteed by the Absorbing Lemma. We remove from H a small set R of vertices, called reservoir see Lemma 6, which has the property, that every k -tuple of V has many neighbours in R. Let H be the remaining hypergraph after removing the vertices from R. Note

4 4 that the property of R allows us to connect every pair P and P 2 of disjoint l-paths in H to one l-path, by connecting the end F end of P with the beginning F beg 2 of P 2 by one edge, where the additional k 2l vertices come from R. We will choose P and R small enough, so that δ k H 2k l +o V H. The third auxiliary lemma, the Path-cover Lemma Lemma 7, asserts that all but on vertices of H can be covered by a family of pairwise disjoint l-paths and, moreover, the number of those paths will be constant independent of n. Consequently, we can connect those paths and P to form an l-cycle by using exclusively vertices from R. This way we obtain an l-cycle in H, which covers all but the on left-over vertices from H and some left-over vertices from R. However, we will ensure that the number of those yet uncovered vertices will be smaller than αn and, hence, we can appeal to the absorption property of P and obtain a Hamilton l-cycle. We now state the Absorbing Lemma, the Reservoir Lemma, and the Path-cover Lemma and give the details of the outline above in Section Auxiliary lemmas. We start with the Absorbing Lemma. This lemma asserts the existence of a relatively short, but powerful l-path P which can absorb any small set U V \ V P. The proof will be carried out in Section 3. Lemma 5 Absorbing Lemma. For all integers k 3 and l < k/2 and every ε > 0 there exists an α > 0 and an n 0 such that for every k-uniform hypergraph H = V, E on V = n n 0 vertices with δ k H εn the following holds. There exists an l-path P H with V P ε 5 n such that for all subsets U V \ V P of size at most U αn and U k ln there exists an l-path Q H with V Q = V P U and, moreover, P and Q have exactly the same ends. The next lemma provides a reservoir R V which we will use to connect short paths to a long one. For a k-uniform hypergraph H = V, E, a subset of the vertices R V and a k -tuple S V k, we denote the set of neighbours of S in R by N R S = {v R \ S : S {v} E} and define deg R S = N R S. Lemma 6 Reservoir Lemma. For every integer k 2 and every reals d, ε > 0 there exists an n 0 such that for every k-uniform hypergraph H = V, E on V = n n 0 vertices with δ k H dn the following holds. There is a set R of size at most εn such that for all k -sets S V k we have degr S dεn/2. Lemma 6 follows directly from the sharp concentration of the hypergeometric distribution. Proof. For given k, d, and ε we choose n 0 sufficiently large and set q = εn. From V q, the set of all subsets of V with size q, we choose a set R uniformly at random. Now let S V k be an arbitrary set of size k and let XS = N R S. Then X S is hypergeometrically distributed with expectation E [X S ] qd 6. Applying Chernoff s inequality for hypergeometric distribution see, e.g., [3, Theorem 2.0] we obtain P [X S dq/2 ] exp dq/30 = exp dεn/30 Thus, with probability n k exp dεn/30 = o every set S V at least dεn/2 neighbours in R. k has

5 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS 5 Finally, we state the Path-cover lemma. By an l-path packing of a k-uniform hypergraph H we mean a family of pairwise vertex disjoint l-paths. Then the Pathcover Lemma asserts that a k-uniform hypergraph H with δ k H 2k l + o V H can be almost perfectly covered by few l-paths. Lemma 7 Path-cover Lemma. For all integers k 3 and l < k/2 and every γ and ε > 0 there exist integers p and n 0 such that for every k-uniform hypergraph H = V, E on V = n n 0 vertices with δ k H 2k l + γ n the following holds. There is an l-path packing of H consisting of at most p paths, which covers all but at most εn vertices of H. The proof of Lemma 7 is based on the weak hypergraph regularity lemma and is given in Section Proof of Theorem 3. In this section we give the proof of the main result, Theorem 3. The proof is based on the three auxiliary lemmas introduced in Section 2.2 and follows the outline given in Section 2.. Proof of Theorem 3. Let integers k 3 and l < k/2 and a real γ > 0 be given. Applying the Absorbing Lemma Lemma 5 for k, l, and ε 5 = γ/4 we obtain α > 0 and n 5. Next we apply the Reservoir Lemma Lemma 6 for k, l, and d = /2k and ε 6 = min{α/2, γ/4} we obtain n 6. Finally, we apply the Path-cover Lemma Lemma 7 with γ 7 = γ/2 and ε 7 = α/2 to obtain p and n 7. For n 0 we choose n 0 = max{n 5, 2n 6, 2n 7, 6p + k 2 /ε 6 }. Now let n n 0, n k ln and let H = V, E be a k-uniform hypergraph on n vertices with δ k H 2k l + γ n. Let P 0 H be the absorbing l-path guaranteed by Lemma 5 applied with k, l, and ε 5. Let F beg 0 and F0 end be the ends of P 0 which we may refer to as sets. Note that V P 0 ε5n 5 < γn/4. Moreover, the path P 0 has the absorption property, i.e. for all U V \ V P 0 with U αn and U k ln l-path Q H s.t. V Q = V P 0 U and Q has the ends F beg 0 and F0 end. be Let V = V \ V P 0 F beg 0 F0 end and let H = H[V ] = V, EH V k the induced subhypergraph of H on V. Note that δ k H 2k l + 3γ/4n V /2k = d V. Due to Lemma 6 we can choose a set R V \ F beg 0 F0 end of size at most ε 6 V ε 6 n such that deg R S ε 6 V /4k F beg 0 F0 end ε 6 n/8k for every S V k. 2 Set V = V \ V P 0 R and let H = H[V ] be the induced subhypergraph of H on V. Clearly, δ k H 2k l + 3γ/4 ε 6 n 2k l + γ/2 V. Consequently, Lemma 7 applied to H with γ 7 and ε 7 yields an l-path packing of H which covers all but at most ε 7 V ε 7 n vertices from V and consists of

6 6 at most p paths. We denote the set of the uncovered vertices in V by T. Further, let P, P 2..., P q with q p denote the l-paths of the packing and let F beg i and Fi end for i =,..., q be the ends of the l-path P i. Recall that the ends of the absorbing l-path P 0 are F beg 0 and F0 end. Note that for each 0 i, j q we have Fi end F beg j = 2l < k. Thus, for any set X R of size k 2l X might be empty we have deg R Fi end F beg j X ε 6 n/8k > p + k due to 2 and the choice of n 0. Consequently, for each i {0,,..., q} we can choose a set Y i R \ 0 j<i Y j such that F end i Y i F beg i+ mod q+ is an edge in EH \ q i=0 EP i. Hence, we can connect all paths P, P 2,..., P q, and P 0 to an l-cycle C H. Let U = V \ V C be the set of vertices not covered by the l-cycle C. Since U R T we have U ε 7 + ε 6 n αn. Moreover, since C is an l-cycle and n k ln we have U k ln. Thus, using the absorption property of P 0 see we can replace the subpath P 0 in C by a path Q since P 0 and Q have the same ends and since V Q = V P 0 U the resulting l-cycle is a Hamilton l-cycle of H. 3. Proof of the Absorbing Lemma In this section we prove Lemma 5, the Absorbing Lemma. Roughly speaking, absorption stands for a local extension of a given structure, which preserves the global structure. For l-paths, e.g., we want to insert a set S of vertices to an existing l-path, i.e. to absorb S, in such a way that the new object is again an l-path which, moreover, has the same ends. Definition 8. Let k 3 and l < k/2 be integers and H = V, E be a k- uniform hypergraph. We say an l-path with three edges P H and ends F beg and F end is an absorbing path for a k l-set S V \V P k l, if there exists an l-path Q with four edges with the same ends F beg and F end and V Q = V P S. Moreover, if P is an absorbing path for S with ends F beg and F end, then we call the t-set T = V P V \S t with t = 3k l + l an absorbing t-tuple for S with ends F beg and F end. Given that an absorbing l-path P for S was part of some long l-path, then the local change of absorbing S does not destroy the long path since the ends of P and Q are the same. Clearly, for any fixed k l-set S there are at most On t absorbing t-tuples. The following proposition, however, says that this bound is achieved up to a constant factor when the minimum k -degree of H is linear in n. Proposition 9. Let k 3, l < k/2, ε > 0, and let H be a k-uniform hypergraph on n 6k/ε vertices with δ k H εn. Then for every k l-set S V k l there are at least ε 5 n t /2 5+3k k 4 absorbing t-tuples T V \S t with t = 3k l + l. We postpone the proof of Proposition 9 and we first deduce Lemma 5 from it. Proof of Lemma 5. Let k 3, l < k/2, and ε > 0 be given. We set t = 3k l + l and fix auxiliary constants ζ = ε5 t 2l! 2 6+3k k 4 t! and ϱ = ζ 6t 2 < ε5 8t.

7 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS 7 Finally we set α = ζϱ/4 and let n 0 6k/ε be sufficiently large. Suppose H = V, E is a k-uniform hypergraph on n n 0 vertices which satisfies δ k H εn. Note that in Proposition 9 the ends of the absorbing t-tuples are not specified yet. This we do retropectively by taking the ends F beg T an arbitrary t-set T V, F T end T of t uniformly at random, i.e. with probability t 2l!/t! a given pair of disjoint, ordered l-tuples will become the ends of T. Hence, due to Proposition 9, the expected number of absorbing t-tuples now with distinguished ends for a fixed k l-set S V k l is at least 2ζ n t. Applying Chernoff s inequality we derive that there is a choice of ends for all t-sets which yields at least ζ n t absorbing t-tuples with distinguished ends for all k l-sets. We fix such a choice and for a fixed k l-set S V k l let T S denote the set of the absorbing t-tuples T for S with ends F beg T and FT end according to this choice. Thus, we have T S ζ n t for all S V k l. Next we pick a family T V t randomly, where each t-tuple T V t is included in T independently with probability p = ϱn/ n t. Hence, we have V E [ T ] = ϱn and E [ T T S ] ζϱn S. k l From Chernoff s inequality we infer that with probability o and T 2ϱn 3 T T S ζϱn/2 for all S V k l. 4 Furthermore, let IT denote the number of intersecting t-tuples in T, i.e. the number of pairs T and T T such that T T. Then n n E [IT ] t p 2 = t2 ϱ 2 n 2 t t n t + 2t2 ϱ 2 n = ζϱn/8 due to the choice of ϱ, and using Markov s inequality we conclude that with probability at least /2 IT ζϱn/4. 5 In particular, the properties 3, 4, and 5 hold simultaneously with positive probability for the randomly chosen family T. So, let T be a family satisfying 3, 4, and 5. By deleting all intersecting t-tuples from T and all those t-tuples which do not absorb any S V k l we obtain a family T T of pairwise disjoint t-tuples of size at most 2ϱn which, due to 4, 5, and the choice of α, satisfies T T S ζϱn/4 = αn 6 for all S V k l. Lastly, we want to connect the t-tuples in T to create an l-path. To this end, and Fi end be the ends of T i. Since every T i with its chosen ends F beg i and Fi end absorbs at least one k l-set, the induced hypergraph H[T i ] must contain an l-path P i with three edges and ends let T = {T,..., T r } for some r 2ϱn and let F beg i

8 8 and Fi end. For i =,..., r observe further that Fi end hence, for any V i of size at least n 4ϱnt and any Y V i F beg i N Vi F end i F beg i+ Y εn 4ϱnt > 0. F beg i+ = 2l and, k 2l we know Thus, we can choose X i N Vi Fi end F beg i+ to connect P i and P i+ through the edge Fi end X i F beg i+. Starting with the set V = V H \ T T V T of size V n 2ϱnt we connect P and P 2. We continue by induction. So suppose for some i < r we chose sets X,..., X i and used them to connect the l-paths P,..., P i to one l-path. With V i = V \ i j= X j which has size at least n 2ϱnt ik 2l > n 4ϱnt and by the observation from above we connect P i and P i+ by choosing X i N Vi Fi end F beg i+. Consequently, we can connect all l-paths P,..., P r to one l-path P containing at most 4ϱnt ε 5 n vertices. Finally, suppose U V \ V P with U αn and U k ln. Then we partition U into q αn/k l pairwise disjoint sets S,..., S q each of size k l. But since 6 holds, we can absorb each S i, i =,..., q one by one taking an unused absorbing t-tuple T i T T S for each S i. This way we obtain an l-path Q which covers exactly the vertices in V P U and the lemma follows. We complete the proof of Lemma 5 by proving Proposition 9. To this end we need the notion of a neighbourhood of a set S V H in a set U V H. This is given by N U S = {X U \ S : S X EH}. Proof of Proposition 9. Let S V k l be an arbitrary set of size k l and set V 0 = V \ S. In the following we will choose pairwise disjoint sets A, B, B 2, C, D, and D 2 whose union forms an absorbing t-tuple for S. We start by choosing A V 0 k 2l arbitrarily. Then the number of choices for A is n k + l. 7 k 2l Set V = V 0 \ A and split S A = Z L Z 2 in an arbitrary way such that L = l and Z = Z 2 = k 2l. We choose B N V Z L and B 2 N V2 Z 2 L where V 2 = V \ B. To compute the number of choices for B and B 2 note that V 2 = n 2k +3l, V 3 = n 2k +2l and for every set X i V i l, i =, 2, we know that deg H Z i L X i εn thus N Vi Z i L X i has size at least εn 2k εn/2, since n 4k/ε. This way we count each possible B i in l ways. Consequently, the number of choices for B and B 2, i.e. N V2 Z L N V3 Z 2 L is at least εn 2 n 2k + 3l n 2k + 2l. 8 2l l l Next, set V 3 = V 2 \ B 2 and for i =, 2 let B i B i of size B i = B i thus, B i may be empty if l =. We choose the set C N V 3 A B B 2. Since V 3 = n 2k + l by arguing as above for B and B 2 we conclude that the number of choices for C is at least εn2 n 2k + lεn 2k Then we set V 4 = V 5 \ C and for C = {v, v 2 }, we choose D N V4 B {v } and with V 5 = V 4 \ D we choose D 2 N V5 B 2 {v 2 }. Note that V 5 = n 2k +

9 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS 9 l 2, V 6 = n 3k and B i {v i } = l +. Thus, again, by arguing as for B, B 2 we derive that the number of choices for D and D 2 is at least 2 εn n 2k + l 2 n 3k. 0 2k l k l 2 k l 2 For given S let and note that T = A B B 2 C D D 2 T = A + B + B 2 + C + D + D 2 = 3k l + l = t. Combining 7, 8, 9, and 0 we obtain that the number of choices for T chosen as above for a given set S is at least ε 5 n k + l ε 5 n ε 5 n 2 7 l 2 k 2 t 2 7+t l 2 k 2 t 2 5+3k k 4. t We now verify that T is indeed an absorbing t-tuple for S. For that we reorder the vertices of T and observe that Note that T = D B {v } A {v 2 } B 2 D 2. E = D B {v }, G = B {v } A {v 2 } B 2, and E 2 = {v 2 } B 2 D 2 are edges in H and form an l-path P with three edges, since E i G = B i {v i} = l, for i =, 2. For the ends of this path we could fix any ordering of any l-set from D i. Moreover, the sets G = B Z L and G 2 = L Z 2 B 2 are also edges of H and E, G, G 2, E 2 forms an l-path Q with V Q = S T, since G i E i = B i = l, for i =, 2 and G G 2 = L = l. The ends of this l-path can be chosen to coincide with the ends of P, since D i G i = for i =, 2. This proves that any set T chosen as above is indeed an absorbing t-tuple for the set S. 4. The Path-cover Lemma In this section we prove the Path-cover Lemma, Lemma 7. The proof combines the techniques in [4] and [9] and relies on the so called weak hypergraph regularity lemma, a straightforward generalisation of Szemerédi s regularity lemma [7] for graphs see e.g. [, 2, 6]. 4.. Almost perfect F k,l -packings. First we show that an n-vertex, k-uniform hypergraph H with minimum degree δ k H n/2k l contains a F k,l -packing which covers all but on vertices of H, where F k,l is defined as follows. Definition 0. For positive integers k and l let F k,l be the k-uniform hypergraph on 2k lk vertices whose vertex set falls into pairwise disjoint sets A, A 2,..., A 2k 2l, B each of size k and whose edge set consists of all sets A i {b} where i [2k 2l ] and b B. Kühn and Osthus [9] considered F 3, -packings, i.e. families of pairwise vertex disjoint copies of F 3,. The proof of the F k,l -packing lemma, Lemma, follows their approach.

10 0 Lemma F k,l -packing Lemma. For all integers k 3 and l < k and every ε > 0 there exists an n 0 such that for every k-uniform hypergraph H = V, E on V = n n 0 vertices the following holds. If deg k S n/2k l for all but at most εn k sets S V k, then H contains a F k,l -packing covering all but at most 5ε /k n vertices. Proof. For given k, l, and ε we choose n 0 large enough. Further set δ = 5ε /k. Suppose A = {F, F 2,..., F i0 } is a largest F k,l -packing leaving the vertex set X V of size X δn uncovered. From the condition on the degree for H we first show the following. Claim 2. There is a family B of size δn/2k k consisting of mutually disjoint k -sets S X k such that degs n/2k l and NX S δn/4k for all S B. Proof. The claim follows from a probabilistic argument. First we split X into two parts X = X X 2 by choosing X 2 X of size X /2k uniformly at random. Thereafter, we take a family S consisting of δn/k k pairwise disjoint sets S X k from X such that degs n/2k l. Such a family exists indeed, since the number of k -sets with degree falling below n/2k l is at most εn k and due to the choice of δ X k εn k k δn k k X. k 2 Next, we claim that at least half, i.e. δn/2k k, of the chosen k -sets S i must satisfy N X S i δn/4k since otherwise the F k,l -packing A was not largest possible. For a contradiction, let S S denote the set of the chosen S i S such that N X S > δn/4k and suppose S = {S,..., S r } has size r δn/2k k. For any k -sets S X k with NX S > X /4k let Y S = N X2 S denote the size of its neighbourhood in X 2. Then Y S has hypergeometric distribution with mean E [Y S ] X /4k /2k δn/8k 2 and applying Chernoff s inequality we conclude p = P [ Y S δn/6k 2 ] exp{ δn/00k 2 }. Thus, with a probability at least X k p = o all sets S X k with N X S > X /4k also satisfy N X2 S n/6k 2. In particular, almost surely N X2 S n/6k 2 is satisfied for all S S and we assume that this indeed happens for the decomposition X = X X 2 we have chosen. Now consider the auxiliary bipartite graph G with vertex classes S and X 2 and with {S, v} being an edge if and only if S {v} H. Then every S has at least δn/6k 2 neighbours, thus, by the well known result of Kövari, Turán, and Sós [7] the graph G contains a K k,k. However, this K k,k in G corresponds to a copy of F k,l in H, which is a contradiction to A being the largest F k,l -packing. Continuing the proof of Lemma, we fix a family B = {S,..., S q }, q = δn/2k k as stated in the claim above. For a set S i B we say that an element F from the F k,l -packing A is good for S i if F contains at least k neighbours of S i, i.e. N V F S i k. With n i denoting the number of good F A for S i and

11 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS t = 2k lk we conclude from the condition on degs i that n 2k l degs δn i k + tn i + δn t 4k δ/2n + tn i. 2 2k l From this we infer that n i δn/8k 3 = n. Next, we want to count all those pairs S, T with T = {F,..., F k } A k such that each F T is good for S B. Such a pair S, T we call a good pair and the number of good pairs is at least B n k δn k /8k 5 k. Thus by averaging we infer that there must be a T and at least δ k n/8k 5 k sets S i B such that S i, T are a good pairs. k Hence, it exists a family B B containing at least δ k n/8k 5 k / 2k lk k pairwise disjoint k -sets S from B and for every j =,... k there exist k vertices v j,..., vj k in F j such that S {v j },..., S {vj k } EH for every S B and j =,... k. Since δ k n/8k 5 k / 2k lk k k 2k l k for sufficiently large n, we can select k families mutually disjoint families {S, i..., S2k 2l i } B for i =,..., k. Now for every i =,..., k the set {Sp i {v j i }: p =,..., 2k 2l, j =,..., k } is the edge set of a copy of F k,l and we obtain k mutually disjoint copies of F k,l this way. Replacing the k -copies F,..., F k by those k copies enlarges the F k,l -packing B, which is a contradiction Weak hypergraph regularity and path embeddings. In this section we introduce the so-called weak hypergraph regularity lemma, a straightforward extension of Szemerédi s regularity lemma [7] for graphs. Further we will find almost perfect path packings in regular k-tuples. Similar results were used by Rödl, Ruciński and Szemerédi in [4] The weak regularity lemma for hypergraphs. Let H = V, E be a k-uniform hypergraph and let A,..., A k be mutually disjoint non-empty subsets of V. We define e H A,..., A k to be the number of edges with one vertex in each A i, i [k] and the density of H with respect to A,..., A k as d H A,..., A k = e HA,..., A k A... A k. We say the k-tuple V,..., V k of mutually disjoint subsets V,..., V k V is ε, d-regular, for constants ε > 0 and d 0, if d H A,..., A k d ε for all k-tuples of subsets A V,..., A k V k satisfying A ε V,..., A k ε V k. We say the k-tuple V,..., V k is ε-regular if it is ε, d-regular for some d 0. The following fact is a direct consequence of the definition above. Fact 3. For an ε, d-regular tuple V,..., V k we have i V,..., V k is ε, d-regular for all ε > ε and ii if for all i [k] the set V i V i has size V i c V i, then V,..., V k is ε/c, d-regular.

12 2 As a straightforward generalisation of the original regularity lemma we obtain the following regularity lemma for graphs see, e.g., [, 2, 6]. Theorem 4 Weak regularity lemma for hypergraphs. For all integers k 2 and t 0, and every ε > 0, there exist T 0 = T 0 k, t 0, ε and n 0 = n 0 k, t 0, ε so that for every k-uniform hypergraph H = V, E on n n 0 vertices, there exists a partition V = V 0 V... V t such that i t 0 t T 0, ii V = V 2 = = V t and V 0 εn, iii for all but at most ε t k sets {i,..., i k } [t] k, the k-tuple Vi,..., V ik is ε-regular. A partition as given in Theorem 4 is called an ε-regular partition of H with lower bound t 0 on the number of vertex classes. Further, we need the notion of the cluster graph. Definition 5. For an ε-regular partition of H and d 0 we refer to the sets V i, i [t] as clusters and define the cluster hypergraph K = Kε, d with vertex set [t] = {, 2,..., t} and {i,..., i k } [t] k being an edge if and only if Vi,..., V ik is ε-regular and dv i,..., V ik d. The following proposition relates the degree condition of H and its cluster hypergraph K. It shows that K almost inherits the minimum degree of H. Proposition 6. Given a k-uniform hypergraph H = V, E with minimum k - degree δ k H 2k l + γ n and an ε-regular partition V = V 0 V... V t with 0 < ε < γ 2 /6 and t 0 8k/ε 3k/γ. Further, let K = Kε, γ/6 be the cluster hypergraph of H. Then the number of k -sets S = {i,..., i k } [t] k violating deg K S 2k l + γ t 4 is at most εt k. Proof. Note first that the cluster hypergraph Kε, γ/6 can be written as the intersection of two hypergraphs D = Dγ/6 and R = Rε both defined on the vertex set [t] and Dγ/6 consists of all sets {i,..., i k } such that dv i,..., V ik γ/6 Rε consists of all sets {i,..., i k } such that V i,..., V ik is ε-regular. Given an arbitrary set S [t] k we first show deg D S 2k l + γ 2 t. 3 To this end note that S = {i,..., i k } represents the tuple V i,..., V ik with n/t m := V ij εn/t for all j [k ]. We consider now the number of edges in H which intersects each V ij in exactly one vertex. From the condition on δ k H this is at least m k 2k l + γ n k m m k 2k l + 2γ n 4 3

13 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS 3 since t t 0 3k/γ. On the other hand, in case 3 does not hold the same number can be bounded from above by 2k l + γ 2 t m k + t γ 6 mk with contradiction to 4. Next, observe that there are at most ε t k < εt k /k sets {i,..., i k } [t] k such that the corresponding tuples V i,..., V ik are not ε-regular, i.e. {i,..., i k } R. Thus, all but at most εt k sets S [t] k satisfy deg R S εt. 5 Since K = D R the proposition follows from 3, 5 and εt γt/ Almost perfect path-packings in regular k-tuples. In this section we show that ε, d-regular k-tuples V,..., V k can be almost perfectly covered by l-paths. Definition 7. Suppose H is a k-uniform, k-partite hypergraph with partition classes V, V 2,..., V k. Then we call an l-path P H with t edges E,..., E t canonical with respect to V, V 2,..., V k if E i E i+ V j or E i E i+ j [l] j [k]\[k l] for all i =, 2,..., t. Further, we say that V i is in end position if it is one of the first or the last l elements in the ordering, i.e. i [l] {k l +,..., k}, whereas V i is in middle position if i {l +,..., k l}. Remark 8. Let t be a odd number. If P with t edges is a canonical path with respect to V,..., V k and n i = V P V i, then { t + /2 if V i is in end position, n i = t if V i is in middle position. The following proposition was essentially proved in [4]. Proposition 9. Suppose H is a k-partite, k-uniform hypergraph with the partition classes V, V 2,..., V k, V i = m for all i [k], and EH dm k. Then there exists a canonical l-path in H with respect to V,... V k with t > dm/2k l edges. Proof. First we consider all possible ends of a canonical l-path P, i.e. all l-sets L V H such that L V i = either for all i [l] or for all i [k] \ [k l]. For a possible end L such that degl = {E H: L E} < dm k l /2 we delete all edges from the current hypergraph which contain L. We keep doing this until every possible end L satisfies degl = 0 or degl dm k l /2 in the present hypergraph. Note that we have deleted less than 2m l dm k l /2 = dm k edges, hence, the final hypergraph H is non-empty. We pick a maximal canonical l-path P H with respect to V,..., V k which has t edges and let the l-set L denote one end of P. Since L is contained in an edge in H we know that degl dm k l /2. On the V j

14 4 other hand, every edge in H which contains L must intersect V P \ L since P is maximal. Thus, we have dm k l t + degl < k 2lt + l m k l k ltm k l. 2 2 This yields t > dm/2k l. We want to use Proposition 9 to cover a ε-regular tuple V,..., V k by l-paths which intersect V,..., V k equally and which, moreover, intersect V k almost as little as possible. Lemma 20. For all integers k 3, l < k/2, and all d, β > 0 there exist ε > 0, integers p and m 0 such that for all m > m 0 the following holds. Suppose V = V, V 2,..., V k is an ε, d-regular k-tuple with V i = 2k 2l m for all i [k ] and V k = k m. Then there is a family consisting of at most p pairwise vertex disjoint l-paths which cover all but at most βm vertices of V. Proof. Let k, l, d, and β be given. We choose ε = min{d/2, β/7k 2, /k!}, p = 2k/ε 2, and m 0 > 2ε 3 sufficiently large. Suppose V = V,..., V k is an ε, d- regular tuple as stated in the lemma. We choose t to be the largest odd number satisfying t ε 2 km/k l and we want to cover V by l-paths each having t edges. To this end, let S k denote the symmetric group and for each permutation τ S k let Vτ = V τ, V τ2,..., V τk, V k. Let p 0 denote the maximal integer for which there exists a family of pairwise disjoint l-paths with exactly t edges each, such that every l-path is canonical with respect to some Vτ, τ S k, and for every τ S k there are either exactly p 0 or p 0 + paths in this family which are canonical with respect to Vτ. Among those families let P p0 be one with maximal cardinality and for each τ S k for which there are p 0 + canonical l-paths with respect to Vτ in P p0 we remove one of those paths to obtain P P p0 with size P = p 0 k!. We will prove that P is the family of l-paths regquired in the lemma. For a family P of paths let V P = P P V P and we claim that there is an r [k] such that V r \ V P p0 < 2kεm. In the opposite case we pick W r V r \ V P p0 with size W r = 2kεm for all r [k] and from regularity of V,..., V k and W r V r we derive that ew,..., W k d ε2kεm k. Since d 2ε it follows from Proposition 9 that for any τ S k there is a canonical l-path with respect to W τ,..., W τk, W k which consists of more than ε 2 km/k l t edges. Note that these l-paths are not necessarily disjoint for different τ. However, we get a contradiction either to the maximality of p 0 or to the maximality of P p0. Thus, with U r = V r V P for all r [k], we derive that there exists an r [k] such that U r V r P p0 \ P t 2kεm V r 3kεm, since P p0 \ P k!, t ε 2 km/k l, and ε /k!. From the above we want to derive that U r V r 7kεm for all r [k] 6

15 DIRAC-TYPE RESULTS FOR LOOSE HAMILTON CYCLES IN HYPERGRAPHS 5 which would imply the lemma, since ε β/7k 2. To this end, note first that canonical l-paths with t edges intersect sets in middle position in exactly t vertices, whereas sets in end positions are intersected in t+/2 vertices see Remark 8. Hence, for all r [k ] we have ] U r = p 0 [k 2lk 2!t + 2l k 2!t + /2 ] = p 0 [2k 2l k 2!t + /2 k 2lk 2! and U k = p 0 k!t + /2. Suppose r k then U r = U r V r 3kεm for all r [k ] and However, this implies U k On the other hand, if r = k then from which we derive 2 p 0 t + 2k 2l k 2!. U r k U r 2k 2l k m 3kεm = V k 3kεm. 2 U k p 0 = t + k! U r 2k 2l m 7kεm = V k 7kεm due to m m 0 2ε 3. In both cases, we obtain 6. Lastly, note that p 0 k!t + /2 V k = k m from which we infer P 2k/ε 2 = p Proof of the Path-cover Lemma. In this section we prove the Lemma 7. Proof of Lemma 7. Given k, l with k > 2l and γ, ε > 0. We apply Lemma 20 with k, l, d = γ/6 and β = ε/3 to obtain ε 20, p 20 and m 20 and subsequently apply Lemma with k, l, ε = ε/3 k /5 to obtain n. Finally, we apply Theorem 4 with k and ε 4 = { γ 2 2 min 6, γ 24k, ε2, ε 20 2k } and { t 4 = max n, 6k } ε 4 to obtain T 4 and n 4. Let p = T 4 p 20 and n 0 max{2k 2 T 4 /ε 4, n 4 } sufficiently large. For a hypergraph H on n n 0 vertices with δ k H 2k l + γn we apply the weak hypergraph regularity lemma Theorem 4 with k, ε 4 and t 4. By possibly moving at most t2k 2l k < ε 4 n vertices to V 0 we obtain an 2ε 4 -regular partition V = V 0 V V 2... V t of H such that the partition classes satisfy V = = V t = 2k 2l k m for some positive integer m. Clearly, V 0 2ε 4 n εn/3 and n/t V i n/2t for all i [t].

16 6 For the k-uniform cluster hypergraph K = K2ε 4, γ/6 of H on the vertex set [t] we know by Proposition 6 that all but at most 2ε 4 t k ε t k of the k - sets S [t] k satisfy deg K S 2k l + γ 4 Thus, by Lemma we find a F k,l -packing in K which covers all but at most 5ε /k t εt/3 vertices of K. Let F be an arbitrary copy of F k,l in the cluster hypergraph K with the vertex set, say, V F = {, 2,..., 2k 2lk } grouped into sets A,..., A 2k 2l, B, all of the same size k. The edges of F are the sets A i {b} with i [2k 2l ] and b B. We will show that the corresponding induced hypergraph H F = H[V V 2... V 2k 2lk ] can be covered by a family of at most 2k 2l k p 20 pairwise disjoint l-paths which leave at most t. 2k 2l k βm 7 vertices of H F uncovered. This would imply that the union of these families for the F k,l -packing contains at most tp 20 p pairwise disjoint l-paths and the number of vertices in H not covered by these l-paths is at most V 0 + εt/3 n/t + tβm εn, as stated in the lemma. To find a family of l-paths satisfying 7 let i [2k 2l ] and by suppressing the dependence on i let a,..., a k be the elements of A i. For each i [2k 2l ] and each a A i we subdivide V a into k pairwise disjoint sets Ua,..., Ua k, each having V a = 2k 2l m k vertices and, subsequently group them into tuples Ua r,..., Ua r k with r [k ]. Moreover, for all b B we subdivide V b into 2k 2l pairwise disjoint sets, each of size V b = k m. 2k 2l Thus, we obtain 2k 2l k such sets and there is a bijection between those sets and the k -tuples Ua r,..., Ua r k. We fix such a bijection arbitrarily and denote the preimage of Ua r,..., Ua r k by Wi r recall that we suppressed the dependence of a,..., a k on i. For each i [2k 2l ] and each b B the set A i {b} forms an edge in K, i.e. the tuple V a,..., V ak, V b is 2ε 4, γ/6-regular. Due to Fact 3 and 2ε 4 ε 20 /2k we derive that the k-tuples Ua r,..., Ua r k, Wi r are all ε 20, γ/6- regular. Hence, for each i [2k 2l ] and each r [k ] we can apply Lemma 20 to Ua r,... Ua r k, Wi r to obtain a family of at most p 20 pairwise disjoint l-paths which cover all but at most βm vertices of Ua r,..., Ua r k, Wi r. Since there are exactly 2k 2l k such k-tuples we obtain at most 2k 2l k p 20 paths in total and the number of vertices in H F not covered by those paths is at most 2k 2l k βm, as stated in 7.

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