On multipartite Hajnal–Szemerédi theorems

Han, Jie; Zhao, Yi

doi:10.1016/j.disc.2013.02.008

Cited by 2 publications

(2 citation statements)

References 30 publications

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“…Since the submission of this paper, Keevash and Mycroft [14] have proved Conjecture 1.2, provided n is large enough. Also, Han and Zhao [10] gave a different proof of Conjecture 1.2 for t = 3, 4, again provided n is large enough.…”

Section: Conjecture 12 (Fischer [6]mentioning

confidence: 99%

A Multipartite Version of the Hajnal–Szemerédi Theorem for Graphs and Hypergraphs

Lo¹,

Markström²

2012

Combinator. Probab. Comp.

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A perfect K t -matching in a graph G is a spanning subgraph consisting of vertex-disjoint copies of K t . A classic theorem of Hajnal and Szemerédi states that if G is a graph of order n with minimum degree δ(G) (t − 1)n/t and t|n, then G contains a perfect K t -matching. Let G be a t-partite graph with vertex classes V 1 , . . . , V t each of size n. We show that, for any γ > 0, if every vertex x ∈ V i is joined to at least (t − 1)/t + γ n vertices of V j for each j = i, then G contains a perfect K t -matching, provided n is large enough. Thus, we verify a conjecture of Fischer [6] asymptotically. Furthermore, we consider a generalization to hypergraphs in terms of the codegree.

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Section: Conjecture 12 (Fischer [6]mentioning

confidence: 99%

A Multipartite Version of the Hajnal–Szemerédi Theorem for Graphs and Hypergraphs

Lo¹,

Markström²

2012

Combinator. Probab. Comp.

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“…Martin and Szemerédi [23] proved a quadripartite version of the Hajnal-Szemerédi Theorem. Han and Zhao [12] reproved the results of [21,23] by using the absorbing method. An approximate version of the multipartite Hajnal-Szemerédi Theorem was given by Csaba and Mydlarz [7].…”

mentioning

confidence: 99%

Tiling Tripartite Graphs with 3-Colorable Graphs: The Extreme Case

Hogenson¹,

Martin²,

Zhao³

2018

Graphs and Combinatorics

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There is a sufficiently large N ∈ hN such that the following holds. If G is a tripartite graph with N vertices in each vertex class such that every vertex is adjacent to at least 2N/3 + 2h− 1 vertices in each of the other classes, then G can be tiled perfectly by copies of K h,h,h . This extends work by two of the authors [Electron. J. Combin, 16 (1), 2009] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that 2N/3 + 2h − 1 in our result can not be replaced by 2N/3 + h − 2 and that if N is divisible by 6h, then we can replace it with the value 2N/3 + h − 1 and this is tight.

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On multipartite Hajnal–Szemerédi theorems

Cited by 2 publications

References 30 publications

A Multipartite Version of the Hajnal–Szemerédi Theorem for Graphs and Hypergraphs

A Multipartite Version of the Hajnal–Szemerédi Theorem for Graphs and Hypergraphs

Tiling Tripartite Graphs with 3-Colorable Graphs: The Extreme Case

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