Packing degenerate graphs

Allen, Peter; Böttcher, Julia; Hladký, Jan; Piguet, Diana

doi:10.1016/j.aim.2019.106739

Cited by 17 publications

(82 citation statements)

References 26 publications

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“…Next we consider any forest F ∈ F η I . By Proposition 6.4 (applied with 2, η, γ playing the roles of c, β, α), for every c ∈ [2] we can choose a subtree T c…”

Section: For All I ∈ [D] and I ∈ [D] The Bipartite Graphsmentioning

confidence: 99%

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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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“…Next we consider any forest F ∈ F η I . By Proposition 6.4 (applied with 2, η, γ playing the roles of c, β, α), for every c ∈ [2] we can choose a subtree T c…”

Section: For All I ∈ [D] and I ∈ [D] The Bipartite Graphsmentioning

confidence: 99%

“…As discussed below, this will be crucial for the current paper.) Very recently, Allen, Böttcher, Hladký and Piguet [2] proved the following result, which allows for packing trees of maximum degree up to o(n/ log n): if H 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Optimal packings of bounded degree trees

et al. 2019

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“…This generalises earlier results of Böttcher, Hladkỳ, Piguet and Taraz on tree packings, as well as results of Messuti, Rödl and Schacht and Ferber, Lee and Mousset on packing separable graphs. Very recently, Allen, Böttcher, Hladkỳ and Piguet were able to show that one can in fact find an approximate decomposition of

K_{n}

into

scriptH

provided that the graphs in

scriptH

have bounded degeneracy and maximum degree

o (n / log n)

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

o (n / log n)

.…”

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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“…Some ideas used in [2] are inspired by the present work. The strongest result on Conjecture 2 for general trees (with no degree restriction) is a result of Montgomery, Pokrovskiy, and Sudakov [18], who proved that 2n + 1 copies of any (n + 1)-vertex tree T pack into K 2n+o(n) .…”

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confidence: 99%

“…E and I(2) E , and let L 1 and L 2 be the corresponding subsets of L. Suppose that c ∈ I(1) E ∪ I (2) E . Then (17) tells us that Q(J, min(I (1) E )) = Q(J, c) = Q(J, min(I (2)E )) = 0 for all but at most 2 ℓ m choices of J ∈ J .…”

mentioning

confidence: 99%

Almost all trees are almost graceful

Adamaszek

Allen

Grosu

et al. 2020

Random Struct Algorithms

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The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1, 2, … , n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each > 0 and for all n > n 0 ( ). Suppose that (i) the maximum degree of T is bounded by O (n∕ log n), and (ii) the vertex labels are chosen from the set {1, 2, … , ⌈(1 + )n⌉}. Then there is an injective labeling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.

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Packing degenerate graphs

Cited by 17 publications

References 26 publications

Optimal packings of bounded degree trees

Optimal packings of bounded degree trees

A bandwidth theorem for approximate decompositions

Almost all trees are almost graceful

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