The Approximate Loebl--Komlós--Sós Conjecture III: : The Finer Structure of LKS Graphs
Authors: Jan Hladký, János Komlós, 
Diana Piguet, Miklós Simonovits, Maya Stein, Endre SzemerédiAuthors Info & Claims
Diana Piguet, Miklós Simonovits, Maya Stein, Endre SzemerédiAuthors Info & ClaimsPages 1017 - 1071
Abstract
This is the third of a series of four papers in which we prove the following relaxation of the Loebl--Komlós--Sós conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the fourth paper, the refined structure will be used for embedding the tree $T$.
References
[1]
P. Erdös, Z. Füredi, M. Loebl, and V. T. Sós, Discrepancy of trees, Studia Sci. Math. Hungar., 30 (1995), pp. 47--57.
[2]
J. Hladký, J. Komlós, D. Piguet, M. Simonovits, M. Stein, and E. Szemerédi, The approximate Loebl--Komlós--Sós Conjecture I: The sparse decomposition, SIAM J. Discrete Math., 31 (2017), pp. 945--982, https://doi.org/10.1137/140982842, https://arxiv.org/abs/1408.3858.
[3]
J. Hladký, J. Komlós, D. Piguet, M. Simonovits, M. Stein, and E. Szemerédi, The approximate Loebl--Komlós--Sós Conjecture II: The rough structure of LKS graphs, SIAM J. Discrete Math., 31 (2017), pp. 983--1016, https://doi.org/10.1137/140982854, https://arxiv.org/abs/1408.3871.
[4]
J. Hladký, J. Komlós, D. Piguet, M. Simonovits, M. Stein, and E. Szemerédi, The approximate Loebl--Komlós--Sós Conjecture III: The finer structure of LKS graphs, SIAM J. Discrete Math., 31 (2017), pp. 1017--1071, https://doi.org/10.1137/140982866, https://arxiv.org/abs/1408.3866.
[5]
J. Hladký, J. Komlós, D. Piguet, M. Simonovits, M. Stein, and E. Szemerédi, The approximate Loebl--Komlós--Sós Conjecture IV: Embedding techniques and the proof of the main result, SIAM J. Discrete Math., 31 (2017), pp. 1072--1148, https://doi.org/10.1137/140982878, https://arxiv.org/abs/1408.3870.
[6]
J. Hladký and D. Piguet, Loebl-Komlós-Sós Conjecture: Dense case, J. Combin. Theory Ser. B, 116 (2016), pp. 123--190.
[7]
J. Hladký, D. Piguet, M. Simonovits, M. Stein, and E. Szemerédi, The approximate Loebl--Komlós--Sós conjecture and embedding trees in sparse graphs, Electron. Res. Announc. Math. Sci., 22, 2015, pp. 1--11.
[8]
D. Piguet and M. J. Stein, An approximate version of the Loebl-Komlós-Sós conjecture, J. Combin. Theory Ser. B, 102 (2012), pp. 102--125.
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DOI:10.1137/sjdmec.31.2
Issue’s Table of Contents© 2017, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2017
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