Loebl–Komlós–Sós Conjecture: Dense case

Hladký, Jan; Piguet, Diana

doi:10.1016/j.jctb.2015.07.004

Cited by 18 publications

(19 citation statements)

References 23 publications

(48 reference statements)

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“…Hence, the trees from F 1 can be embedded into C 1 , by using Lemma 5.4, as before, with t := | F 1 | and t 1 , t 2 chosen appropriately. Indeed, inequalities (17) and (11) ensure that condition (iii) of the lemma holds. Furthermore, because of (13) and (18), we know that…”

Section: The Proof Of Theorem 13mentioning

confidence: 98%

“…For the case k = n 2 , Ajtai, Komlós and Szemerédi [1] proved an approximate version for large n, and years later Zhao [25] proved the exact result for large n.An approximate version of the Loebl-Komlós-Sós conjecture for dense graphs was proved by Piguet and Stein [19]. The exact version for dense graphs was settled by Piguet and Hladký [17], and independently by Cooley [6]. For sparse graphs, Hladký, Komlós, Piguet, Szemerédi and Stein proved an approximate version of the Loebl-Komlós-Sós conjecture in a series of four papers [13][14][15][16].Maximum and minimum degree.…”

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confidence: 99%

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Degree Conditions for Embedding Trees

Besomi¹,

Pavez-Signé²,

Stein³

2019

SIAM J. Discrete Math.

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We propose the following conjecture: For every fixed α ∈ [0, 1 3 ), each graph of minimum degree at least (1 + α) k 2 and maximum degree at least 2(1 − α)k contains each tree with k edges as a subgraph. Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asymptotically exceeding k 2 and 4 3 k, respectively, as long as the host graph avoids a specific structure.Let us give a quick outline of the most relevant directions that have been suggested in the literature.Minimum degree. It is very easy to see that every graph of minimum degree at least k contains each tree with k edges, and this is sharp (consider the disjoint union of complete graphs of order k).Average degree. The famous Erdős-Sós conjecture from 1964 (see [7]) states that every graph with average degree strictly greater than k − 1 contains each tree with k edges. If true, the conjecture is sharp.This conjecture has received a lot of attention over the last three decades, in particular, a proof was announced by Ajtai, Komlós, Simonovits and Szemerédi in the early 1990's. Many particular cases have been settled since then, see e.g. [4,5,10,12,22,23].Median degree. The Loebl-Komlós-Sós conjecture from 1992 (see [8]) states that every graph of median degree at least k contains each tree with k edges. If true, also this conjecture is sharp. For the case k = n 2 , Ajtai, Komlós and Szemerédi [1] proved an approximate version for large n, and years later Zhao [25] proved the exact result for large n.An approximate version of the Loebl-Komlós-Sós conjecture for dense graphs was proved by Piguet and Stein [19]. The exact version for dense graphs was settled by Piguet and Hladký [17], and independently by Cooley [6]. For sparse graphs, Hladký, Komlós, Piguet, Szemerédi and Stein proved an approximate version of the Loebl-Komlós-Sós conjecture in a series of four papers [13][14][15][16].Maximum and minimum degree. A new angle to the tree containment problem was introduced in 2016 by Havet, Reed, Stein, and Wood [11]. They impose bounds on both the minimum and the maximum degree. More precisely, they suggest that every graph of minimum degree at least 2k 3 and maximum degree at least k contains each tree with k edges. Again, this is sharp if true. We call this conjecture the 2 3 -conjecture, for progress see [11,20,21].In [3], the present authors proposed a variation of this approach, conjecturing that every graph of minimum degree at least k 2 and maximum degree at least 2k contains each tree with k edges. We call this the 2k-k 2 conjecture. An example illustrating the sharpness of the conjecture, and a version for trees with maximum degree bounded by k 1 67 and large dense host graph can be found in [3].New conjecture. Comparing the two varian...

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Section: The Proof Of Theorem 13mentioning

confidence: 98%

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confidence: 99%

Degree Conditions for Embedding Trees

Besomi¹,

Pavez-Signé²,

Stein³

2019

SIAM J. Discrete Math.

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“…The proof of Lemma 6.10 is standard, and is given for example in [HP16,Lemma 5.12]. Lemma 6.10 suggests the following definitions.…”

Section: Embedding Small Treesmentioning

confidence: 99%

The Approximate Loebl--Komlós--Sós Conjecture IV: Embedding Techniques and the Proof of the Main Result

Hladký¹,

Komlós²,

Piguet³

et al. 2017

SIAM J. Discrete Math.

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This is the last paper of a series of four papers in which 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.In the first two papers of this series, we decomposed the host graph G, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlós-Sós Conjecture contains one of ten specific configurations. In this paper we embed the tree T in each of the ten configurations.Mathematics Subject Classification: 05C35 (primary), 05C05 (secondary).

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“…The conjecture has been solved exactly for large dense graphs [Coo09,HP16] and proved to be asymptotically true for sparse graphs [HKP + 17a, HKP + 17b, HKP + 17c, HKP + 17d] (see [HPS + 15] for an overview).…”

Section: Introductionmentioning

confidence: 99%