Covering and tiling hypergraphs with tight cycles

Abstract

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.

Cover 3-uniform hypergraphs by vertex-disjoint tight paths

“…The next lemma was proved in [10], which states that for regular slices J as in Theorem 4.4, the codegree conditions are also preserved by R d (G). Note that its original version allows G to be (µ, θ)-dense as well.…”

Dirac-type results for tilings and coverings in ordered graphs

“…Proposition 1.9 implies that, asymptotically, the minimum degree threshold for ensuring an Hcover in a graph G is the same as the minimum degree threshold for ensuring a single copy of H in G. In [22,Theorem 5], Kühn, Osthus and Treglown asymptotically determined the Ore-type degree condition that forces an H-cover for any fixed graph H. There has also been several recent papers concerning minimum ℓ-degree conditions that force H-covers in k-uniform hypergraphs; see, e.g., [11,12,14].…”

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

“…and C (3) 5 . Han, Lo and Sanhueza-Matamala [29] determined c r−1 (C (r) t ) for all r 3 and t > 2r 2 . In this paper we investigate c 1 (n, F) and c 1 (F) for various 3-graphs F. We first consider K…”

“…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).…”