Covering and tiling hypergraphs with tight cycles

Abstract

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.

Exact Minimum Codegree Thresholds for $K_4^-$-Covering and $K_5^-$-Covering

“…For general kgraphs, the function c i (n, F ) was determined for some special families of k-graphs F . For example, Han, Lo, and Sanhueza-Matamala [3]…”

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).…”