Minimum Vertex Degree Threshold for ‐tiling*

Abstract

Abstract. We prove that the vertex degree threshold for tiling C 3 4 (the 3-uniform hypergraph with four vertices and two triples) in a 3-uniform hypergraph on n ∈ 4N vertices is otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.

Minimum codegree threshold for Hamilton ℓ-cycles in k-uniform hypergraphs

“…The exact value of t 2 (n, Y 3,2 ) was determined recently by Czygrinow, DeBiasio, and Nagle [2]. We [10] determined t 1 (n, Y 3,2 ) very recently. The key lemma in our proof, Lemma 2.8, shows that every k-graph H on n vertices with δ k−1 (H) ≥ ( 1 2k−b − o(1))n either contains an almost perfect Y k,b -tiling or is in the extremal case.…”

Minimum vertex degree thresholds for tiling complete 3-partite 3-graphs

“…In this paper, we investigate the minimum vertex degree conditions for tiling complete 3-partite 3-graphs K. Our result is best possible, up to the error term γn 2 . We remark that in some cases (e.g., K = K 1,1,t for t ≥ 2) it seems possible to remove the error term and obtain exact results -this was done for K 1,1,2 in [4,15]. In general, in order to obtain an exact result, we need to have a stability version of the almost tiling lemma and a stability version of the absorbing lemma, together with an analysis of the 3-graphs that look like extremal examples.…”

Packing k-partite k-uniform hypergraphs

“…For perfect Hpackings other than a perfect matching, results are much more sparse. Lo and Markström [27] found the asymptotic values of δ 1 (K 3 3 (m), n) and δ 1 (K 4 4 (m), n), where δ 1 (H, n) denotes the smallest integer δ such that any k-graph G on n vertices with deg G ({x}) ≥ δ for any x ∈ V (G) contains a perfect H-packing, and K r r (m) denotes the complete r-partite r-graph (defined PACKING k-PARTITE k-UNIFORM HYPERGRAPHS 3 below) whose vertex classes each have size m. More recently, Han and Zhao [12] gave the exact value of δ 1 (K 3 4 −2e, n) for large n, whilst Lenz and Mubayi [24] proved that for any linear k-graph F (meaning that any two edges of F intersect in at most one vertex), any sufficiently large 'quasirandom' k-graph with linear density contains a perfect F -packing. However, in general our knowledge of conditions which guarantee a perfect H-packing in a k-graph G remains very limited.…”