Packing tight Hamilton cycles in uniform hypergraphs

Journal of Combinatorics, 2016

A k-uniform hypergraph H contains a Hamilton ℓ-cycle, if there is a cyclic ordering of the vertices of H such that the edges of the cycle are segments of length k in this ordering and any two consecutive edges f i , f i+1 share exactly ℓ vertices. We consider problems about packing and counting Hamilton ℓ-cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A ⊆ V (H), we denote by d H (A) the number of distinct edges f ∈ E(H) for which A ⊆ f , and set δ d (H) to be the minimum d H (A) over all A ⊆ V (H) of size d. We show that if a k-uniform hypergraph on n vertices H satisfies δ k−1 (H) ≥ αn for some α > 1/2, then for every ℓ < k/2 H contains (1 − o(1)) n • n! • α ℓ!(k−2ℓ)! n k−ℓ Hamilton ℓ-cycles. The exponent above is easily seen to be optimal. In addition, we show that if δ k−1 (H) ≥ αn for α > 1/2, then H contains f (α)n edge-disjoint Hamilton ℓ-cycles for an explicit function f (α) > 0. For the case where every (k − 1)-tuple X ⊂ V (H) satisfies d H (X) ∈ (α ± o(1))n, we show that H contains edge-disjoint Haimlton ℓ-cycles which cover all but o (|E(H)|) edges of H. As a tool we prove the following result which might be of independent interest: For a bipartite graph G with both parts of size n, with minimum degree at least δn, where δ > 1/2, and for p = ω(log n/n) the following holds. If G contains an r-factor for r = Θ(n), then by retaining edges of G with probability p independently at random, w.h.p the resulting graph contains a (1 − o(1))rp-factor.

Journal of Combinatorial Theory, Series B, 2010

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 − 1)-degree, i.e. the minimum number of edges containing a fixed set of k − 1 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 − 1)-degree (1/2 + o(1))n contains such a tight Hamilton cycle. We study the similar question for Hamilton-cycles. A Hamilton-cycle in an n-vertex, k-uniform hypergraph (1 ≤ < 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 vertices. We prove sufficient minimum (k − 1)-degree conditions for Hamiltoncycles if < k/2. In particular, we show that for every < k/2 every n-vertex, k-uniform hypergraph with minimum (k − 1)-degree (1/(2(k −)) + o(1))n contains such a loose Hamilton-cycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus (for = 1), 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.

Combinatorics, Probability and Computing, 2017

A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph H k n,p has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈ ( $\binom{[n]}{k}$ ) to E with probability p, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in H k n,p is $p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$ the best known bounds for the packing problem are around p = polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: for p ≥ log C n/n k−1, a ra...

Journal of Combinatorial Theory, Series B, 2022

For any even integer k ≥ 6, integer d such that k/2 ≤ d ≤ k−1, and sufficiently large n ∈ (k/2)N, we find a tight minimum d-degree condition that guarantees the existence of a Hamilton (k/2)-cycle in every k-uniform hypergraph on n vertices. When n ∈ kN, the degree condition coincides with the one for the existence of perfect matchings provided by Rödl, Ruciński and Szemerédi (for d = k − 1) and Treglown and Zhao (for d ≥ k/2), and thus our result strengthens theirs in this case.

The Electronic Journal of Combinatorics, 2015

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K&gt;0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for g...

Electronic Notes in Discrete Mathematics, 2011

We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ 1 pHq ě`7 16`o p1q˘`n 2˘c ontains a loose Hamilton cycle. This bound is asymptotically best possible.

Random Structures &amp; Algorithms, 2021

We investigate the emergence of subgraphs in sparse pseudo‐random k‐uniform hypergraphs, using the following comparatively weak notion of pseudo‐randomness. A k‐uniform hypergraph H on n vertices is called ‐pseudo‐random if for all (not necessarily disjoint) vertex subsets with we have urn:x-wiley:rsa:media:rsa21052:rsa21052-math-0004For any linear k‐uniform F, we provide a bound on in terms of and F, such that (under natural divisibility assumptions on n) any k‐uniform ‐pseudo‐random n‐vertex hypergraph H with a mild minimum vertex degree condition contains an F‐factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo‐random hypergraphs and, along the way, we also derive conditions which guarantee the appearance of any fixed sized subgraph. All results imply corresponding bounds for stronger notions of hypergraph pseudo‐randomness such as jumbledness or large spectral gap. As a consequence, ‐pseudo‐random k‐graphs as above cont...

Combinatorics, Probability and Computing, 2009

Random Structures &amp; Algorithms, 2018

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1,…,Ct, each of which is oriented arbitrarily, a digraph ∼(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ∼(n, p) w.h.p. contains (1 ± o(1))n!pn copies of C, provided .

2018

Gyárfás, Győri and Simonovits [3] proved that if a 3-uniform hypergraph with n vertices has no linear cycles, then its independence number α ≥ 2n 5 . The hypergraph consisting of vertex disjoint copies of a complete hypergraph K 5 on five vertices shows that equality can hold. They asked whether this bound can be improved if we exclude K 5 as a subhypergraph and whether such a hypergraph is 2-colorable. In this paper, we answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph doesn’t contain K 5 as a subhypergraph, then it is 2-colorable. This result clearly implies that its independence number α ≥ ⌈n 2 ⌉. We show that this bound is sharp. Gyárfás, Győri and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n− 2 when n ≥ 10.