Covering and tiling hypergraphs with tight cycles

Source abstract

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/(2 s ) + 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.