A multipartite Hajnal–Szemerédi theorem

Keevash, Peter; Mycroft, Richard

doi:10.1016/j.jctb.2015.04.003

Cited by 39 publications

(38 citation statements)

References 9 publications

(20 reference statements)

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“…Since the proof of Theorem 1.1 there have been many generalizations obtained in several directions. For example, Kühn and Osthus [38] characterized, up to an additive constant, the minimum degree which ensures that a graph G contains a perfect H-tiling for an arbitrary graph H. Keevash and Mycroft [27] proved an analogue of the Hajnal-Szemerédi theorem in the setting of r-partite graphs, whilst there are now several generalizations of Theorem 1.1 in the setting of directed graphs (see e.g., [11,12,48]). See [37] for a survey including many of the results on graph tiling.…”

Section: Perfect Tilings In Graphsmentioning

confidence: 99%

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

Han

Morris

Treglown

2020

Random Struct Algorithms

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A perfect Kr‐tiling in a graph G is a collection of vertex‐disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr‐tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed 0<α<1−1/r we determine how many random edges one must add to an n‐vertex graph G of minimum degree δ(G)≥αn to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr‐tiling. As one increases α we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best‐possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, α=0) and that of Hajnal and Szemerédi [18] (which demonstrates that for α≥1−1/r the initial graph already houses the desired perfect Kr‐tiling).

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Section: Perfect Tilings In Graphsmentioning

confidence: 99%

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

Han

Morris

Treglown

2020

Random Struct Algorithms

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“…Although the Hajnal‐Szemerédi Theorem was proved many years ago, there has been significant recent activity on related theorems. For example, Alon‐Yuster , Komlós‐Sárközy‐Szemerédi and Kühn‐Osthus have all proved theorems similar to the Hajnal‐Szemerédi Theorem where complete graphs factors are replaced with H ‐factors where H is an arbitrary graph; Kierstead‐Kostochka proved the Hajnal‐Szemerédi Theorem with an Ore‐type degree condition ; Fischer , Martin‐Szemerédi , and Keevash‐Mycroft have proved multipartite variants; and Wang , Keevash‐Sudakov , Czygrinow‐Kierstead‐Molla , Czygrinow‐DeBiasio‐Kierstead‐Molla , Treglown and Balogh‐Lo‐Molla have all proved analogues of the Hajnal‐Szemerédi Theorem in directed and oriented graphs.…”

Section: Introductionmentioning

confidence: 99%

Triangle factors of graphs without large independent sets and of weighted graphs

Balogh

Molla

Sharifzadeh

2016

Random Struct Algorithms

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The classical Corrádi‐Hajnal theorem claims that every n‐vertex graph G with δ(G)≥2n/3 contains a triangle factor, when 3|n. In this paper we present two related results that both use the absorbing technique of Rödl, Ruciński and Szemerédi. Our main result determines the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n‐vertex graph with α(G)=o(n) and δ(G)≥(1/2+o(1))n, then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratically many edges, and δ(G)≥(1/2+o(1))n, then G has a Kr‐factor for n sufficiently large. We also propose many related open problems whose solutions could show a relationship with Ramsey‐Turán theory. Additionally, we also consider a fractional variant of the Corrádi‐Hajnal Theorem, settling a conjecture of Balogh‐Kemkes‐Lee‐Young. Let t∈(0,1) and w:E(Kn)→[0,1]. We call a triangle t‐heavy if the sum of the weights on its edges is more than 3t. We prove that if 3|n and w is such that for every vertex v the sum of w(e) over edges e incident to v is at least (1+2t3+o(1))n, then there are n/3 vertex disjoint heavy triangles in G. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 669–693, 2016

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“…A factor of transversals is a set of pairwise-disjoint transversals covering all vertices. The problem of finding sufficient conditions for the existence of independent transversals and factors of independent transversals in k-partite graphs was studied by several researchers [1,2,3,4,5,6,7,8,10,12,13,14,15,16,17] not least because it is strongly related to the Hajnal-Szemerédi Theorem and to concepts such as the strong chromatic number and list coloring.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the modified Fischer's Conjecture implies Conjecture 1. The modified Fischer's Conjecture has been solved for k = 3 and large n by Magyar and Martin [13], for k = 4 and large n by Martin and Szemerédi [14] and, finally, for every fixed k and sufficiently large n = n(k) by Keevash and Mycroft [10]. However, as all of these proofs require n to be very large compared to k, they do not imply Conjecture 1 nor an upper bound close to the conjecture.…”

Section: Introductionmentioning

confidence: 99%

On Factors of Independent Transversals in $k$-Partite Graphs

Yuster¹

2021

Electron. J. Combin.

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A $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite graph induced by any pair of parts is a matching. An independent transversal in such a graph is an independent set that intersects each part in a single vertex. A factor of independent transversals is a set of $n$ pairwise-disjoint independent transversals. Let $f(k)$ be the smallest integer $n_0$ such that every $[k,n,1]$-graph has a factor of independent transversals assuming $n \geqslant n_0$. Several known conjectures imply that for $k \geqslant 2$, $f(k)=k$ if $k$ is even and $f(k)=k+1$ if $k$ is odd. While a simple greedy algorithm based on iterating Hall's Theorem shows that $f(k) \leqslant 2k-2$, no better bound is known and in fact, there are instances showing that the bound $2k-2$ is tight for the greedy algorithm. Here we significantly improve upon the greedy algorithm bound and prove that $f(k) \leqslant 1.78k$ for all $k$ sufficiently large, answering a question of MacKeigan.

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A multipartite Hajnal–Szemerédi theorem

Cited by 39 publications

References 9 publications

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

Triangle factors of graphs without large independent sets and of weighted graphs

On Factors of Independent Transversals in $k$-Partite Graphs

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