On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

Hàn, Hip; Person, Yury; Schacht, Mathias

doi:10.1137/080729657

Cited by 119 publications

(158 citation statements)

References 8 publications

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“…However, as pointed out in [3], the conditions on G in the case r = 3 and ℓ = 2 in Theorem 1.2 are not strong enough to guarantee this property. Consequently, the absorber construction from [12,22] fails. Moreover, this turns out to be the case for an arbitrary r and ℓ: Consider a graph G obtained by taking an r-partite complete graph with vertex classes V 1 , .…”

Section: Proof Strategymentioning

confidence: 99%

On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

Nenadov¹,

Pehova²

2020

SIAM J. Discrete Math.

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A seminal result of Hajnal and Szemerédi states that if a graph G with n vertices has minimum degree δ(G) ≥ (r − 1)n/r for some integer r ≥ 2, then G contains a K r -factor, assuming r divides n. Extremal examples which show optimality of the bound on δ(G) are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Túran theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results:• For any r > ℓ ≥ 2, if G is a graph satisfying δ(G) ≥ r−ℓ r−ℓ+1 n+Ω(n) and α ℓ (G) = o(n), that is, a largest K ℓ -free induced subgraph has at most o(n) vertices, then G contains a K r -factor. This is optimal for ℓ = r − 1 and extends a result of Balogh, Molla, and Sharifzadeh who considered the case r = 3.• If a graph G satisfies δ(G) = Ω(n) and α * r (G) = o(n), that is, every induced K r -free r-partite subgraph of G has at least one vertex class of size o(n), then it contains a K r -factor. A similar statement is proven for a general graph

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Section: Proof Strategymentioning

confidence: 99%

On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

Nenadov¹,

Pehova²

2020

SIAM J. Discrete Math.

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“…In this paper we consider vertex degrees in 3-graphs. Hàn, Person and Schacht [4] showed that m 1 (3, n) = 5 9 + o(1) n 2 .…”

Section: Introductionmentioning

confidence: 99%

Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

Zhang

Zhao

2018

Electron. J. Combin.

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We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-graph without isolated vertex. More precisely, suppose that H is a 3-uniform hypergraph whose order n is sufficiently large and divisible by 3. If H contains no isolated vertex and deg(u)+ deg(v) > 2 3 n 2 − 8 3 n + 2 for any two vertices u and v that are contained in some edge of H, then H contains a perfect matching. This bound is tight.

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“…We know that 4|U Following (7), we have 2 i=1 deg(u i ) + deg(v 1 ) < 3sn − 3η 1 n(n − 3η 1 n) + n/3 − 3. Since ε ≪ η 1 and n is sufficiently large, this contradicts (5).…”

Section: Proof Of Theoremmentioning

confidence: 92%

Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

Zhang

Zhao

2019

Electron. J. Combin.

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Let n, s be positive integers such that n is sufficiently large and s ≤ n/3. Suppose H is a 3-uniform hypergraph of order n. If H contains no isolated vertex and deg(u) + deg(v) > 2(s − 1)(n − 1) for any two vertices u and v that are contained in some edge of H, then H contains a matching of size s. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when s = n/3. n−1 2 − n−s 2 and n ≥ 3s, then H contains a matching of size s.

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On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

Cited by 119 publications

References 8 publications

On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

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