Proof of the Loebl–Komlós–Sós conjecture for large, dense graphs

Cooley, Oliver

doi:10.1016/j.disc.2009.05.030

Cited by 27 publications

(26 citation statements)

References 6 publications

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“…The proofs of Theorem 1.3 in both [1] and [3] are similar to that of Zhao in [5] in the special case k = n/2, although substantial extra difficulties arise in the more general case.…”

Section: Conjecture 12 (Loebl Komlós Sós)mentioning

confidence: 88%

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Loebl-Komlós-Sós Conjecture: dense case

Cooley

Hladký

Piguet

2009

Electronic Notes in Discrete Mathematics

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“…The proofs of Theorem 1.3 in both [1] and [3] are similar to that of Zhao in [5] in the special case k = n/2, although substantial extra difficulties arise in the more general case.…”

Section: Conjecture 12 (Loebl Komlós Sós)mentioning

confidence: 88%

“…This was proved independently in [1] and [3]. Theorem 1.3 For any q > 0 there exists an integer n 0 = n 0 (q) such that for any n > n 0 and k > qn the following holds: If G is a graph on n vertices with at least n/2 vertices of degree k, then T k+1 ⊆ G.…”

Section: Conjecture 12 (Loebl Komlós Sós)mentioning

confidence: 89%

Loebl-Komlós-Sós Conjecture: dense case

Cooley

Hladký

Piguet

2009

Electronic Notes in Discrete Mathematics

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“…We postpone a detailed discussion of similarities between our approach and theirs and of our own contribution until Section 2. After the first version of this manuscript was posted on the arXiv, Oliver Cooley [5] published an independent proof of Theorem 1.5.…”

Section: Theorem 13mentioning

confidence: 99%

Loebl–Komlós–Sós Conjecture: Dense case

Hladký

Piguet

2016

Journal of Combinatorial Theory, Series B

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“…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%

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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Proof of the Loebl–Komlós–Sós conjecture for large, dense graphs

Cited by 27 publications

References 6 publications

Loebl-Komlós-Sós Conjecture: dense case

Loebl-Komlós-Sós Conjecture: dense case

Loebl–Komlós–Sós Conjecture: Dense case

Degree Conditions for Embedding Trees

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