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Abstract
A kuniform 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 kuniform 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 kpartite kuniform hypergraph, which may be of independent interest. For hypergraphs F and H, a perfect Ftiling in H is a spanning collection of vertexdisjoint copies of F. For k ≥ 3, there are currently only a handful of known Ftiling results when F is kuniform but not kpartite. If s 6≡ 0 mod k, then C k s is not kpartite. Here we prove an Ftiling result for a family of non kpartite kuniform hypergraphs F. Namely, for s ≥ 5k 2 , every kuniform 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.
Original language  English 

Pages (fromto)  136 
Journal  Combinatorics, Probability and Computing 
Volume  0 
DOIs  
Publication status  Published  13 Oct 2020 
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Dive into the research topics of 'Covering and tiling hypergraphs with tight cycles'. Together they form a unique fingerprint.Projects
 1 Finished

A graph theoretical approach for combinatorial designs
Engineering & Physical Science Research Council
1/11/16 → 31/10/18
Project: Research Councils