Degrees giving independent edges in a hypergraph

Daykin, D. E.; Häggkvist, Roland

doi:10.1017/s0004972700006924

Cited by 47 publications

(66 citation statements)

References 5 publications

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“…(The case d = 1 of this is already due to Daykin and Häggkvist [6].) These bounds were slightly improved by Markström and Ruciński [22], using similar techniques, to…”

Section: 2mentioning

confidence: 92%

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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Abstract. A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k2 . As a consequence of our result, we extend bounds of Bollobás, Daykin and Erdős by asymptotically determining the minimum vertex degree which forces a matching of size t < 0.48n/(k − 1) in a k-uniform hypergraph on n vertices. We also obtain further results on d-degrees which force large matchings. In addition we improve bounds of Markström and Ruciński on the minimum ddegree which forces a perfect matching in a k-uniform hypergraph on n vertices. Our approach is to inductively prove fractional versions of the above results and then translate these into integer versions.

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“…(The case d = 1 of this is already due to Daykin and Häggkvist [6].) These bounds were slightly improved by Markström and Ruciński [22], using similar techniques, to…”

Section: 2mentioning

confidence: 92%

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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“…when H = K k k consists of k vertices and one edge). The first bounds for the latter case were given by Daykin and Häggkvist [7], and later Rödl, Ruciński and Szemerédi [31] showed that n/2 − k ≤ δ(K k k , n) ≤ n/2 for all sufficiently large n (indeed, they actually determined the exact value of δ(K k k , n) for large values of n). Beyond this, the value of δ(H, n) is known for three other 3-graphs, all on four vertices.…”

Section: Perfect Packings In Graphsmentioning

confidence: 97%

Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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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.

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“…Now, fix any triple (j 1 , j 2 , j 3 ) and any K ′ which was assigned to it. Let x be the vertex of K ′ in Y i q , and consider paths xx 1 x 2 x 3 y of length 4 in G with x ℓ ∈ X ′i j ℓ for ℓ ∈ [3] and y ∈ X ′i q ∩ v∈V (K ′ )\{x} N (v). Since every vertex of K ′ \ {x} is good and does not lie in W i , at most kβn vertices y ∈ X ′i q fail to be adjacent to all of V (K ′ ) \ {x}.…”

Section: Recall That This Means Thatmentioning

confidence: 99%

A multipartite Hajnal–Szemerédi theorem

Keevash

Mycroft

2015

Journal of Combinatorial Theory, Series B

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Abstract. The celebrated Hajnal-Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K k -packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.

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Degrees giving independent edges in a hypergraph

Abstract: For r-partite and for r-uniform hypergraphs bounds are given for the minimum degree which ensures d independent edges.

Cited by 47 publications

References 5 publications

Fractional and integer matchings in uniform hypergraphs

Fractional and integer matchings in uniform hypergraphs

Packing k-partite k-uniform hypergraphs

A multipartite Hajnal–Szemerédi theorem

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