A blow-up lemma for approximate decompositions

Kim, Jaehoon; Kühn, Daniela; Osthus, Deryk; Tyomkyn, Mykhaylo

doi:10.1090/tran/7411

Cited by 43 publications

(83 citation statements)

References 43 publications

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“…This was improved in [2] by allowing the graphs to be packed to be spanning. The paper [5] proves a near-perfect packing result for families of graphs with bounded maximum degree which are otherwise unrestricted. Both Tree Packing Conjectures for trees of bounded maximum degree were solved in [4].…”

mentioning

confidence: 83%

Packing degenerate graphs greedily

Allen

Böttcher

Hladký

et al. 2017

Electronic Notes in Discrete Mathematics

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mentioning

confidence: 83%

Packing degenerate graphs greedily

Allen

Böttcher

Hladký

et al. 2017

Electronic Notes in Discrete Mathematics

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“…We then aim to pack each

H_{t}

into

G_{t}

. As described below, for each

H \in {scriptH}_{t}

, most of the edges will be embedded via the blow‐up lemma for approximate decompositions proved in .…”

Section: Outline Of the Argumentmentioning

confidence: 99%

“…This was recently proved by Joos, Kim, Kühn and Osthus for the case where

n

is large and each

T_{i}

has bounded degree. The crucial tool for this was the blow‐up lemma for approximate decompositions of

ε

‐regular graphs

G

by Kim, Kühn, Osthus and Tyomkyn . In particular, this lemma implies that if

scriptH

is a family of bounded degree

n

‐vertex graphs with

e false(scriptH false) ⩽ false(1 - o (1) false) (\frac{0pt}{n})

, then

K_{n}

has an approximate decomposition into

scriptH

.…”

Section: Introductionmentioning

confidence: 99%

“…Our main tool in the proof of Theorem will be the recent blow‐up lemma for approximate decompositions by Kim, Kühn, Osthus and Tyomkyn : roughly speaking, given a set

scriptH

n

‐vertex bounded degree graphs and an

n

‐vertex graph

G

with

e (H) ⩽ (1 - o (1)) e (G)

consisting of super‐regular pairs, it guarantees a packing of

scriptH

G

(such super‐regular pairs arise from applications of Szemerédi's regularity lemma). Theorem gives the precise statement of the special case that we shall apply (note that the original blow‐up lemma of Komlós, Sárközy and Szemerédi corresponds to the case where

scriptH

consists of a single graph).…”

Section: Introductionmentioning

confidence: 99%

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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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“…Write

C : = 1 \dots c

. To show (

†

), we will use a variant of the Blow‐up lemma . First, we do some pre‐processing.…”

Section: Sketch Of the Proof Of Theoremmentioning

confidence: 99%

Proof of Komlós's conjecture on Hamiltonian subsets

Kim

Liu

Sharifzadeh

et al. 2017

Proc. London Math. Soc.

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Abstract. Komlós conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph K d+1 minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when d is sufficiently large. In fact we prove a stronger result: for large d, any graph G with average degree at least d contains almost twice as many Hamiltonian subsets as K d+1 , unless G is isomorphic to K d+1 or a certain other graph which we specify.

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A blow-up lemma for approximate decompositions

Cited by 43 publications

References 43 publications

Packing degenerate graphs greedily

Packing degenerate graphs greedily

A bandwidth theorem for approximate decompositions

Proof of Komlós's conjecture on Hamiltonian subsets

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