Packing trees of unbounded degrees in random graphs

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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“…This implies an approximate version of the tree packing conjecture when the trees have maximum degree

o (n / log n)

. The latter improves a bound of Ferber and Samotij which follows from their work on packing (spanning) trees in random graphs.…”

Section: Introductionsupporting

confidence: 52%

A bandwidth theorem for approximate decompositions

Condon

Kim

Kühn

et al. 2018

Proc. London Math. Soc.

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“…Generalising in the direction of removing the restriction to bounded degree graphs, Ferber and Samotij [11] showed two near-perfect packing results for trees, one for spanning trees of maximum degree O n 1/6 log −6 n , and one for almost spanning trees of maximum degree O n/ log n . The latter result also follows in the particular case of Ringel's Conjecture from the work of Adamaszek, Allen, Grosu, Hladký [1].…”

Section: Introductionmentioning

confidence: 99%

Packing degenerate graphs

Allen

Böttcher

Hladký³

et al. 2019

Advances in Mathematics

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“…For

k = 2

there are few overlapps between the matchings and it requires a bit more careful treatment. For an example illustrating how to deal with it, the reader is referred to .…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…Then, whp in

H_{k n, p}^{k}

the number of edge‐disjoint perfect matchings,

t

, satisfies

t = (1 + o (1)) true(\begin{matrix} k n - 1 \\ k - 1 \end{matrix} true) p

.Remark We would like to give the following remarks: The case

k = 2

is a bit more complicated to handle using our technique, and in fact better tools are known for this case (generalizations of Hall's Theorem for finding “many” edge‐disjoint perfect matchings). For a non‐trivial example of applying the “online sprinkling” technique for graphs, the reader is referred to . Our

p

is optimal up to a

p o l y l o g (n)

factor as it can be easily seen that for

p = o (\log n / n^{k - 1})

, a typical

H_{k n, p}^{k}

contains isolated vertices and therefore has no perfect matchings. Moreover, our value

t

is asymptotically optimal as well.…”

Section: Introductionmentioning

confidence: 99%

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Packing perfect matchings in random hypergraphs

Ferber

2017

Random Struct Algorithms

Self Cite

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We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer k≥3, the binomial k‐uniform random hypergraph Hn,pk contains N:=(1−o(1))true(n−1k−1true)p edge‐disjoint perfect matchings, provided p≥log⁡Cnnk−1, where C:=C(k) is an integer depending only on k. Our result for N is asymptotically optimal and for p is optimal up to the polylog(n) factor. This significantly improves a result of Frieze and Krivelevich.

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Packing trees of unbounded degrees in random graphs

Cited by 13 publications

References 24 publications

A bandwidth theorem for approximate decompositions

A bandwidth theorem for approximate decompositions

Packing degenerate graphs

Packing perfect matchings in random hypergraphs

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