TY - JOUR

T1 - Covering and tiling hypergraphs with tight cycles

AU - Han, Jie

AU - Lo, Allan

AU - Sanhueza-Matamala, Nicolás

N1 - Publisher Copyright:
© 2020 The Author(s)

PY - 2021/3

Y1 - 2021/3

N2 - A k-uniform tight cycle 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 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. 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, 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 is not k-partite. Here we prove an F-Tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for, every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect-Tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

AB - A k-uniform tight cycle 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 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. 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, 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 is not k-partite. Here we prove an F-Tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for, every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect-Tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

UR - http://www.scopus.com/inward/record.url?scp=85094818097&partnerID=8YFLogxK

U2 - 10.1017/S0963548320000449

DO - 10.1017/S0963548320000449

M3 - Article

AN - SCOPUS:85094818097

SN - 0963-5483

VL - 30

SP - 288

EP - 329

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 2

ER -