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.

中文翻译：

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用紧密循环覆盖和平铺超图

一种ķ- 均匀的紧密循环$C_s^k$是一个超图s>ķ具有循环排序的顶点，使得每个ķ在此排序下的连续顶点形成一条边。这对 （ķ,s) 如果 gcd (ķ,s) = 1 或ķ/gcd (ķ,s） 甚至。我们证明如果$s \ge 2{k^2}$和H是一个ķ-至少具有最小共度的均匀超图（1/2 +○(1))|五(H)|，那么每个顶点都被$C_s^k$. 如果 (ķ,s) 是可接受的。我们的主要工具允许我们任意重新排列紧凑路径环绕完整路径的顺序ķ- 部分ķ-统一的超图，可能是独立的兴趣。对于超图F和H, 一个完美的F- 平铺H是顶点不相交副本的跨越集合F. 为了$k \ge 3$, 目前只有少数已知的F- 平铺结果时F是ķ-统一但不统一ķ- 部分。如果s≢ 0 模组ķ， 然后$C_s^k$不是ķ- 部分。这里我们证明一个F- 非家庭的平铺结果ķ- 部分ķ- 均匀超图F. 即，对于$s \ge 5{k^2}$， 每一个ķ- 均匀超图H最小同度数至少 (1/2 + 1/(2s) +○(1))|V(H)| 有一个完美的$C_s^k$-平铺。此外，如果ķ是偶数和 (ķ,s) 是可接受的。