Loebl-Komlós-Sós Conjecture: dense case

Cooley, Oliver; Hladký, Jan; Piguet, Diana

doi:10.1016/j.endm.2009.07.103

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Cited by 4 publications

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Embedding trees in graphs with independence number two

Chen

2018

Discrete Math. Algorithm. Appl.

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Let [Formula: see text] be a graph of order [Formula: see text] and [Formula: see text] an integer not great than [Formula: see text]. Erdős and Sós conjectured that if [Formula: see text] has at least [Formula: see text] edges, then [Formula: see text] contains all trees of order [Formula: see text]. Loebl, Komlós and Sós conjectured that if at least [Formula: see text] vertices of [Formula: see text] have degree at least [Formula: see text], then [Formula: see text] contains all trees of order [Formula: see text]. In this paper, it is shown that both the conjectures are true for graphs with independence number two, and our results generalize some known results.

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Embedding trees in graphs with independence number two

Chen

2018

Discrete Math. Algorithm. Appl.

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The Loebl–Komlós–Sós Conjecture for Trees of Diameter $5$ and for Certain Caterpillars

Piguet¹,

Stein²

2008

Electron. J. Combin.

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The approximate Loebl-Komlós-Sós conjecture and embedding trees in sparse graphs

Hladký¹,

Piguet²,

Simonovits³

et al. 2015

ERA-MS

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Loebl, Komlós and Sós conjectured that every n-vertex graph G with at least n/2 vertices of degree at least k contains each tree T of order k + 1 as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of k.For our proof, we use a structural decomposition which can be seen as an analogue of Szemerédi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of G to embed a given tree T .The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].

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Loebl-Komlós-Sós Conjecture: dense case

Abstract: We prove a version of the Loebl-Komlós-Sós Conjecture for large dense graphs. For any q > 0 there exists n 0 ∈ N such that for any n > n 0 and k > qn the following holds: If G has median degree at least k, then any tree of order at most k + 1 is a subgraph of G.

Cited by 4 publications

References 2 publications

Embedding trees in graphs with independence number two

Embedding trees in graphs with independence number two

The Loebl–Komlós–Sós Conjecture for Trees of Diameter $5$ and for Certain Caterpillars

The approximate Loebl-Komlós-Sós conjecture and embedding trees in sparse graphs

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Loebl-Komlós-Sós Conjecture: dense case

Abstract: We prove a version of the Loebl-Komlós-Sós Conjecture for large dense graphs. For any q > 0 there exists n 0 ∈ N such that for any n > n 0 and k > qn the following holds: If G has median degree at least k, then any tree of order at most k + 1 is a subgraph of G.

Cited by 4 publications

References 2 publications

Embedding trees in graphs with independence number two

Embedding trees in graphs with independence number two

The Loebl&ndash;Komlós&ndash;Sós Conjecture for Trees of Diameter $5$ and for Certain Caterpillars

The approximate Loebl-Komlós-Sós conjecture and embedding trees in sparse graphs

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The Loebl–Komlós–Sós Conjecture for Trees of Diameter $5$ and for Certain Caterpillars