On the decomposition threshold of a given graph

Glock, Stefan; Kühn, Daniela; Lo, Allan; Montgomery, Richard; Osthus, Deryk

doi:10.48550/arxiv.1603.04724

Cited by 10 publications

(31 citation statements)

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“…Recent developments. Since submission of the original manuscript, there have been a number of further developments: Firstly, Glock, Kühn, Montgomery, Lo and Osthus [10] obtained further results on the decomposition threshold of graphs which strengthen Theorem 1.2. These imply e.g.…”

Section: Introduction and Resultsmentioning

confidence: 96%

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Fractional clique decompositions of dense graphs and hypergraphs

Barber

Kühn

et al. 2017

Journal of Combinatorial Theory, Series B

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Our main result is that every graph G on n ≥ 10 4 r 3 vertices with minimum degree δ(G) ≥ (1−1/10 4 r 3/2 )n has a fractional Kr-decomposition. Combining this result with recent work of Barber, Kühn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F -decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant c k > 0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least (1 − c k /r 2k−1 )n has a fractional Kis the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large F -divisible complete graph has an F -decomposition.

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Section: Introduction and Resultsmentioning

confidence: 96%

“…a minimum degree of (δ * Kr + o(1))n guarantees a K r -decomposition. Also, [10] determines the decomposition threshold for bipartite graphs.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Fractional clique decompositions of dense graphs and hypergraphs

Barber

Kühn

et al. 2017

Journal of Combinatorial Theory, Series B

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“…Barber, Kühn, Lo, and Osthus studied perfect packings of triangles or even cycles into graphs of large minimum degree [6]. Packings of arbitrary small graphs into graphs of large minimum degree were considered by Glock, Kühn, Lo, Montgomery, and Osthus in [14]. A wellknown conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n − 1)/2 directed Hamilton cycles.…”

Section: Introductionmentioning

confidence: 99%

Packing spanning graphs from separable families

2017

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Let G be a separable family of graphs. Then for all positive constants ǫ and ∆ and for every sufficiently large integer n, every sequence G 1 , . . . , G t ∈ G of graphs of order n and maximum degree at most ∆ such that e(G 1 )+· · ·+e(G t ) ≤ (1−ǫ) n 2 packs into K n . This improves results of Böttcher, Hladký, Piguet, and Taraz when G is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most ∆.The proof uses the local resilience of random graphs and a special multi-stage packing procedure.

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“…k . The value of δ 0+ k has been subject to much attention recently: one reason is that by results of [5,19], for k ≥ 3 the approximate decomposition threshold δ 0+ k is equal to the analogous threshold δ dec k which ensures a 'full' K kdecomposition of any n-vertex graph G with δ(G) ≥ (δ dec k + o(1))n which satisfies the necessary divisibility conditions. A beautiful conjecture (due to Nash-Williams in the triangle case and Gustavsson in the general case) would imply that δ dec k = 1 − 1/(k + 1) for k ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

“…• We conjecture that the error term νe(G) in condition (iii) of Theorem 1.2 can be improved. Note that it cannot be completely removed unless one assumes some divisibility conditions on G. However, even additional divisibility conditions will not always ensure a 'full' decomposition under the current degree conditions: indeed, for C 4 , the minimum degree threshold which guarantees a C 4 -decomposition of a graph G is close to 2n/3, and the extremal example is close to regular (see [5] for details, more generally, the decomposition threshold of an arbitrary bipartite graph is determined in [19]). • It would be interesting to know whether the condition on separability can be omitted.…”

Section: Introductionmentioning

confidence: 99%

A bandwidth theorem for approximate decompositions

Condon

Kim

Kühn

et al. 2018

Proc. London Math. Soc.

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We provide a degree condition on a regular n‐vertex graph G which ensures the existence of a near optimal packing of any family scriptH of bounded degree n‐vertex k‐chromatic separable graphs into G. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Böttcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δk be the infimum over all δ⩾1/2 ensuring an approximate Kk‐decomposition of any sufficiently large regular n‐vertex graph G of degree at least δn. Now suppose that G is an n‐vertex graph which is close to r‐regular for some r⩾(δk+ofalse(1false))n and suppose that H1,⋯,Ht is a sequence of bounded degree n‐vertex k‐chromatic separable graphs with 0true∑ie(Hi)⩽(1−ofalse(1false))e(G). We show that there is an edge‐disjoint packing of H1,⋯,Ht into G. If the Hi are bipartite, then r⩾(1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs G of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.

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On the decomposition threshold of a given graph

Cited by 10 publications

References 0 publications

Fractional clique decompositions of dense graphs and hypergraphs

Fractional clique decompositions of dense graphs and hypergraphs

Packing spanning graphs from separable families

A bandwidth theorem for approximate decompositions

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