Given 3≤k≤s, we say that a k-uniform hypergraph Csk is a
tight cycle on s vertices if there is a cyclic ordering of the vertices of
Csk such that every k consecutive vertices under this ordering form an
edge. We prove that if k≥3 and s≥2k2, then every k-uniform
hypergraph on n vertices with minimum codegree at least (1/2+o(1))n has
the property that every vertex is covered by a copy of Csk. Our result is
asymptotically best possible for infinitely many pairs of s and k, e.g.
when s and k are coprime.
A perfect Csk-tiling is a spanning collection of vertex-disjoint copies
of Csk. When s is divisible by k, the problem of determining the
minimum codegree that guarantees a perfect Csk-tiling was solved by a
result of Mycroft. We prove that if k≥3 and s≥5k2 is not divisible
by k and s divides n, then every k-uniform hypergraph on n vertices
with minimum codegree at least (1/2+1/(2s)+o(1))n has a perfect
Csk-tiling. Again our result is asymptotically best possible for infinitely
many pairs of s and k, e.g. when s and k are coprime with k even.Comment: Revised version, accepted for publication in Combin. Probab. Compu