Large Bounded Degree Trees in Expanding Graphs

Balogh, József; Csaba, Béla; Pei, Martin; Samotij, Wojciech

doi:10.37236/278

Cited by 39 publications

(51 citation statements)

References 15 publications

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“…To extend T 1 to the almost spanning tree T ′ , we will use the following corollary of a tree embedding result of Haxell (this is her Theorem 1 with ℓ = 1 and each d i = Δ). We note that it was first observed by Balogh, Csaba, Pei, and Samotij that this is applicable in sparse random graphs. For a graph G and vertex set X ⊆ V ( G ), we let

N_{G} false(X false) : = \cup_{x \in X} N_{G} false(x false)

.…”

Section: Auxiliary Lemmasmentioning

confidence: 59%

“…For almost spanning trees it was shown by Alon, Krivelevich and Sudakov that, for some constant C = C ( ε ,Δ), the random graph G ( n , C / n ) alone a.a.s. contains any tree with at most (1 − ε ) n vertices and maximum degree at most Δ, where the bounds on C = C ( ε ,Δ) have subsequently been improved . Since the random graph G ( n , C / n ) a.a.s.…”

Section: Introductionmentioning

confidence: 99%

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Universality for bounded degree spanning trees in randomly perturbed graphs

Böttcher

Han

Kohayakawa

et al. 2019

Random Struct Algorithms

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We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph Gα on n vertices with δpGαq ě αn for α ą 0 and we add to it the binomial random graph Gpn, C{nq, then with high probability the graph Gα Y Gpn, C{nq contains copies of all spanning trees with maximum degree at most ∆ simultaneously, where C depends only on α and ∆.

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N_{G} false(X false) : = \cup_{x \in X} N_{G} false(x false)

.…”

Section: Auxiliary Lemmasmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Universality for bounded degree spanning trees in randomly perturbed graphs

Böttcher

Han

Kohayakawa

et al. 2019

Random Struct Algorithms

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“…Theorem 1 (contagious sets in spectral expanders) follows from a proof similar to that of Theorem 2, using a result of [11] that shows that every vertex of an (n, d, λ)-graph is a root of a sufficiently large tree (the minimal degree of a nonleaf node in the tree degree gets smaller when λ approaches d from below, hence our bounds deteriorate as λ grows). The resulting algorithm is random-parallel rather than random-sequential because there is no need to use Corollary 12 -we can use Lemma 10 instead.…”

Section: Lemma 11mentioning

confidence: 98%

Contagious Sets in Expanders

Coja-Oghlan

Feige

Krivelevich

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least r active neighbors, where r > 1 is the activation threshold. A contagious set is a set whose activation results with the entire graph being active. Given a graph G, let m(G, r) be the minimal size of a contagious set. It is known that for every d-regular or). We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs.The general flavor of our results is that sufficiently strong expansion properties imply that m(G, 2) ≤ O( In addition, we demonstrate that rather weak assumptions on the girth and/or the spectral gap suffice in order to imply that m(G, 2) ≤ O(. For example, we show this for graphs of girth at least 7, and for graphs with λ(G) < (1 − )d, provided the graph has no 4-cycles.Our results are algorithmic, entailing simple and efficient algorithms for selecting contagious sets. * Goethe University. acoghlan@math.uni-frankfurt.de.

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“…Indeed, Alon, Krivelevich, and Sudakov [AKS07] proved that G n, C n (for a suitable C = C(ε, ∆)) a.a.s. contains all trees of order (1 − ε)n with maximum degree at most ∆ (the constant C was later improved in [BCPS10]).…”

Section: Related Tree Containment Problemsmentioning

confidence: 99%

The Approximate Loebl--Komlós--Sós Conjecture I: The Sparse Decomposition

Hladký¹,

Komlós

Piguet³

et al. 2017

SIAM J. Discrete Math.

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In a series of four papers we prove the following relaxation of the Loebl-Komlós-Sós Conjecture: For every α > 0 there exists a number k 0 such that for every k > k 0 every n-vertex graph G with at least ( 1 2 + α)n vertices of degree at least (1 + α)k contains each tree T of order k as a subgraph.The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: we decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree.Mathematics Subject Classification: 05C35 (primary), 05C05 (secondary).

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Large Bounded Degree Trees in Expanding Graphs

Cited by 39 publications

References 15 publications

Universality for bounded degree spanning trees in randomly perturbed graphs

Universality for bounded degree spanning trees in randomly perturbed graphs

Contagious Sets in Expanders

The Approximate Loebl--Komlós--Sós Conjecture I: The Sparse Decomposition

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