Tight Co‐Degree Condition for Packing of Loose Cycles in 3‐Graphs

Abstract. Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpiński, Ruciński and Szymańska; Szymańska previously showed that this problem is NP-hard for a minimum codegree of n/k − cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.

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“…Then n ∪ i is a linear combination of vectors in L µ 6 Q ∪ (H i ), so (C2) holds. Also, (C1) holds as…”

Section: By (A4) and (A7) So Proposition 54 Implies Thatmentioning

confidence: 97%

“…There is a large literature on minimum degree conditions for perfect matchings in hypergraphs, see e.g. [1,2,6,7,10,15,17,18,19,20,22,23,24,25,26,28,30,31,32] and the survey by Rödl and Ruciński [27] for details.…”

Section: Introductionmentioning

confidence: 99%

Polynomial-time perfect matchings in dense hypergraphs

Keevash

Knox

Mycroft

2015

Advances in Mathematics

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“…There are some previously known results for tiling problems regarding -cycles. 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].…”

Section: Tiling Thresholdsmentioning

confidence: 99%

Covering and tiling hypergraphs with tight cycles

Han

Sanhueza‐Matamala

2017

Electronic Notes in Discrete Mathematics

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A k-uniform tight cycle C k s 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 ≥ 2k 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 k s . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order of 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 ≥ 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 k s 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 ≥ 5k 2 , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V (H)| has a perfect C k s -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

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“…the set of z ∈ V \ {x, y} such that xyz ∈ E(G). The minimum codegree of G is δ 2 (G) = min xy∈V (2) d(x, y). The link graph of a vertex x ∈ V (G) is the collection G x of all pairs uv such that xuv ∈ E(G).…”

Section: Notationmentioning

confidence: 99%