Packing k-partite k-uniform hypergraphs

Mycroft, Richard

doi:10.1016/j.jcta.2015.09.007

Cited by 22 publications

(6 citation statements)

References 35 publications

(66 reference statements)

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“…. , V t } such that |e ∩ V i | 1 for all edges e ∈ E and all 1 i t. A (k, t)-graph H is complete if E consists of all k-sets e such that |e ∩ V i | 1, for all 1 i t. Recently, Mycroft [24] determined the asymptotic value of t(n, K) for all complete (k, k)-graphs K. However, much less is known for non-k-partite k-graphs. For more results on tiling thresholds for k-graphs, see the survey of Zhao [29].…”

Section: Tiling Thresholdsmentioning

confidence: 99%

“…Whenever C is a 3-uniform loose cycle, t(n, C) was determined exactly by Czygrinow [6]. For general loose cycles C in k-graphs, t(n, C) was determined asymptotically by Mycroft [24] and exactly by Gao, Han and Zhao [12]. For tight cycles C k s with s ≡ 0 mod k, Mycroft [24] proved that t(n, C k s ) = (1/2 + o(1))n. Note that all mentioned cycle tiling results correspond to cases where the cycles are k-partite (since k-uniform loose cycles are k-partite for k 3).…”

Section: Theorem 12 (Erdős [10]mentioning

confidence: 99%

“…For general loose cycles C in k-graphs, t(n, C) was determined asymptotically by Mycroft [24] and exactly by Gao, Han and Zhao [12]. For tight cycles C k s with s ≡ 0 mod k, Mycroft [24] proved that t(n, C k s ) = (1/2 + o(1))n. Note that all mentioned cycle tiling results correspond to cases where the cycles are k-partite (since k-uniform loose cycles are k-partite for k 3). In contrast, we will investigate the covering and tiling problems for the tight cycle C k s in some cases where C k s is not necessarily a (k, k)-graph.…”

Section: Theorem 12 (Erdős [10]mentioning

confidence: 99%

“…On the other hand, recall that the case s ≡ 0 mod k was solved asymptotically by Mycroft [24], so we study the complementary case. We prove an upper bound on t(n, C k s ) which is valid whenever s ≡ 0 mod k and s 5k 2 .…”

Section: Proposition 13 Let 2 K < S N With N Divisible By S Then Tmentioning

confidence: 99%

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Covering and tiling hypergraphs with tight cycles

Han

Sanhueza‐Matamala

2020

Combinator. Probab. Comp.

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A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$ . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest. For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$ , there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$ , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$ -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

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Section: Tiling Thresholdsmentioning

confidence: 99%

Section: Theorem 12 (Erdős [10]mentioning

confidence: 99%

Section: Theorem 12 (Erdős [10]mentioning

confidence: 99%

Section: Proposition 13 Let 2 K < S N With N Divisible By S Then Tmentioning

confidence: 99%

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Covering and tiling hypergraphs with tight cycles

Han

Sanhueza‐Matamala

2020

Combinator. Probab. Comp.

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“…Beyond perfect matchings, codegree tiling thresholds have now been determined for a number of small 3-graphs, including K (3) 4 [35,45], K (3)− 4 [30,43] and K (3)−− 4 (K (3) 4 with two edges removed) [10,39]. In addition, the codegree tiling thresholds for r-partite r-graphs have been studied recently [9,25,26,29,52] For minimum vertex-degree tiling thresholds, fewer results are known. The vertex-degree thresholds for perfect matchings were determined for 3-graphs by Han, Person and Schacht [28] (asymptotically) and by Kühn, Osthus and Treglown [41] and Khan [38] (exactly).…”

mentioning

confidence: 99%

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

Falgas–Ravry

Markström

Zhao

2020

Combinator. Probab. Comp.

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We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

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Disjoint perfect matchings in 3‐uniform hypergraphs

Zhang

2017

Journal of Graph Theory

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For a hypergraph H, let δ1false(Hfalse) denote the minimum vertex degree in H. Kühn, Osthus, and Treglown proved that, for any sufficiently large integer n with n≡0(mod3), if H is a 3‐uniform hypergraph with order n and δ1false(Hfalse)>0ptn−12−0pt2n/32 then H has a perfect matching, and this bound on δ1false(Hfalse) is best possible. In this article, we show that under the same conditions, H contains at least ⌈(2n+3)/9⌉ pairwise disjoint perfect matchings, and this bound is sharp.

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Packing k-partite k-uniform hypergraphs

Cited by 22 publications

References 35 publications

Covering and tiling hypergraphs with tight cycles

Covering and tiling hypergraphs with tight cycles

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

Disjoint perfect matchings in 3‐uniform hypergraphs

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