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… Show more
“…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.…”
Section: Perfect Packings In Graphsmentioning
confidence: 69%
“…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.…”
Section: Perfect Packings In Graphsmentioning
confidence: 99%
“…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).…”
Abstract. Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let δ(H, n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G) ≥ δ contains a perfect H-packing. The problem of determining δ(H, n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ≥ 3. Here we determine the asymptotic value of δ(H, n) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rödl and Ruciński on the value of δ(H, n) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.
“…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.…”
Section: Perfect Packings In Graphsmentioning
confidence: 69%
“…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.…”
Section: Perfect Packings In Graphsmentioning
confidence: 99%
“…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).…”
Abstract. Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let δ(H, n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G) ≥ δ contains a perfect H-packing. The problem of determining δ(H, n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ≥ 3. Here we determine the asymptotic value of δ(H, n) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rödl and Ruciński on the value of δ(H, n) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.
“…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.…”
Section: 3mentioning
confidence: 94%
“…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.…”
Abstract. Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpiński, Ruciński and Szymańska; Szymańska previously showed that this problem is NP-hard for a minimum codegree of n/k − cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.
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