$$F$$ F -Factors in Hypergraphs Via Absorption

Abstract

Given integers n ≥ k > l ≥ 1 and a k-graph F with |V (F )| divisible by n, define t k l (n, F ) to be the smallest integer d such that every kgraph H of order n with minimum l-degree δ l (H) ≥ d contains an F -factor. A classical theorem of Hajnal and Szemerédi [9] implies that t 2 1 (n, K t ) = (1 − 1/t)n for integers t. For k ≥ 3, t k k−1 (n, K k k ) (the δ k−1 (H) threshold for perfect matchings) has been determined by Kühn and Osthus [17] (asymptotically) and Rödl, Ruciński and Szemerédi [24] (exactly) for large n.In this paper, we generalise the absorption technique of Rödl, Ruciński and Szemerédi [24] to F -factors. We determine the asymptotic values of t k 1 (n, K k k (m)) for k = 3, 4 and m ≥ 1. In addition, we show that for t > k = 3 and γ > 0, t 3 2 (n, K 3 t ) ≤ 1 − 2 * Lemma 1.1 (Absorption lemma for F -factors). Let t and i be positive integers and let η > 0. Let F be a hypergraph of order t. Then, there is an integer n 0 = n 0 (t, i, η) satisfying the following: Suppose that H is an (F, i, η)-closed hypergraph of order n ≥ n 0 . Then there exists a vertex subset U ⊆ V (H) of size |U | ≤ (η/2) t n/(4it(t − 1)) with |U | ∈ tZ such that there exists an F -factor in H[U ∪ W ] for every vertex set W ⊆ V \ U of size |W | ≤ (η/2) 2t n/(32i 2 t(t − 1) 2 ) with |W | ∈ tZ.Note that in the above lemma H and F are not necessarily kgraphs, but we only consider k-graphs in this paper. When we say that H has an almost F -factor T , we mean that T is a set of vertexdisjoint copies of F in H such that |V (H) \ V (T )| < ε|V (H)| for some small ε > 0. Equipped with the absorption lemma, we can break down the task of finding an F -factor in large hypergraphs H into the following algorithm.Algorithm for finding F -factors. Remove a small set T 1 of vertex-disjoint copies of F from H such that the resultant graph Hfor some integer i and constant η > 0. 2. Find a vertex set U ⊆ V (H 1 ) satisfying the conditions of the absorption lemma. Set H 2 = H 1 [V (H 1 ) \ U ]. 3. Show that H 2 contains an almost F -factor, i.e. a set T 2 of vertexdisjoint copies of F such that |V (H 2 ) \ V (T 2 )| < ε|V (H 2 )| for small ε > 0.

Packing k-partite k-uniform hypergraphs

“…Theorem 1.1 could also be proved by the so-called 'absorbing method' by using similar arguments and results to those of Lo and Markström [27]. However, our methods also give stronger bounds for many k-partite k-graphs K, for this we make the following definitions.…”

“…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.…”

“…The value of δ(K 3 4 − 2e, n) was found to be n/4 + o(n) by Kühn and Osthus [21]; recently Czygrinow, DeBiasio and Nagle [4] found the exact value for large n to be either n/4 or n/4 + 1 according to the parity of n/4. Lo and Markström [25,27] showed that δ(K 3 4 − e, n) = n/2 + o(n) and that δ(K 3 4 , n) = 3n/4 + o(n). Simultaneously with the latter, Keevash and Mycroft [16] showed that the exact value of δ(K 3 4 , n) for large n is 3n/4 − 1 or 3n/4 − 2, again according to the parity of n/4; these results confirmed a conjecture of Pikhurko [29], who had previously shown that δ(K 3 4 , n) ≤ 0.8603n, and who gave the construction which establishes the lower bound on δ(K 3 4 , n).…”

Polynomial-time perfect matchings in dense hypergraphs

“…The conjecture of Hàn, Person and Schacht mentioned above would therefore imply all cases of Conjecture 10.6 with (1 − 1/k) k−ℓ ≤ 1/2, and some cases of the latter conjecture are implied by partial results for the former. Specifically, Conjecture 10.6 holds in the case k = 3, ℓ = 1 by a result of Hàn, Person and Schacht [10], in the case k = 4, ℓ = 1 by a result of Lo and Markström [24], and in the cases k = 5, ℓ = 1 and k = 6, ℓ = 2 by results of Alon, Frankl, Huang, Rödl, Ruciński and Sudakov [2]. To our knowledge all other cases remain open.…”

“…There is a large literature on minimum degree conditions for perfect matchings in hypergraphs, see e.g. [1,2,6,7,10,15,17,18,19,20,22,23,24,25,26,28,30,31,32] and the survey by Rödl and Ruciński [27] for details.…”