Optimal path and cycle decompositions of dense quasirandom graphs

Glock, Stefan; Kühn, Daniela; Osthus, Deryk

doi:10.1016/j.jctb.2016.01.004

Cited by 19 publications

(16 citation statements)

References 24 publications

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“…In [13], this result is used as a tool to prove several decomposition and packing conjectures involving Hamilton cycles and perfect matchings. Also, in [22], it is used to derive optimal decomposition results for dense quasi-random graphs into other structures, including linear forests. A different generalization of Walecki's theorem is given by the Alspach conjecture, which states that for odd n, the complete graph K n should have a decomposition into cycles C 1 , .…”

Section: 3mentioning

confidence: 99%

A blow-up lemma for approximate decompositions

Kim

Kühn

Osthus

et al. 2018

Trans. Amer. Math. Soc.

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Abstract. We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let G be a quasi-random n-vertex graph and suppose H1, . . . , Hs are bounded degree n-vertex graphs with s i=1 e(Hi) ≤ (1 − o(1))e(G). Then H1, . . . , Hs can be packed edge-disjointly into G. The case when G is the complete graph Kn implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem.We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi's regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy and Szemerédi to the setting of approximate decompositions.

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Section: 3mentioning

confidence: 99%

A blow-up lemma for approximate decompositions

Kim

Kühn

Osthus

et al. 2018

Trans. Amer. Math. Soc.

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“…The main idea for the proof is based on that of Theorem 4.1 in [18]. We will use results from [27], [28] and [29] which imply that robustly expanding graphs enjoy very strong Hamiltonicity properties.…”

Section: Approximate Cycle Decompositionmentioning

confidence: 99%

“…We can then obtain a Hamilton decomposition of the remaining graph via Theorem 4.2. The argument builds on ideas from [18].…”

Section: Approximate Cycle Decompositionmentioning

confidence: 99%

Optimal packings of bounded degree trees

et al. 2019

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We prove that if T1, . . . , Tn is a sequence of bounded degree trees so that Ti has i vertices, then Kn has a decomposition into T1, . . . , Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

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“…For values of d of the form Θ(n), Kühn and Osthus proved in [25] that every 'quasi-random' regular graph has a Hamilton cycle decomposition, and hence, a 1-factorization. Moreover, Glock, Kühn and Osthus [16] also studied optimal edge-colorings in the dense quasi-random case when the underlying graph is not necessarily regular. Usually, the main problem with handling values of d which grow with n is that the so-called 'configuration model' (see [4] for more details) is not very useful in this regime.…”

Section: Random Graphsmentioning

confidence: 99%

1‐Factorizations of pseudorandom graphs

Ferber

Jain

2020

Random Struct Algorithms

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A 1-factorization of a graph G is a collection of edge-disjoint perfect matchings whose union is E(G). In this paper, we prove that for any > 0, an (n, d, λ)-graph G admits a 1-factorization provided that n is even, C 0 ≤ d ≤ n − 1 (where C 0 = C 0 ( ) is a constant depending only on ), and λ ≤ d 1− . In particular, since (as is well known) a typical random d-regular graph G n,d is such a graph, we obtain the existence of a 1-factorization in a typical G n,d for all C 0 ≤ d ≤ n−1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G, which is better by a factor of 2 nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.

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Optimal path and cycle decompositions of dense quasirandom graphs

Cited by 19 publications

References 24 publications

A blow-up lemma for approximate decompositions

A blow-up lemma for approximate decompositions

Optimal packings of bounded degree trees

1‐Factorizations of pseudorandom graphs

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