The approximate Loebl-Komlós-Sós conjecture I: The sparse decomposition

Hladký, J. and Komlós, János and Piguet, D. and Simonovits, Miklós and Stein, M. and Szemerédi, Endre (2017) The approximate Loebl-Komlós-Sós conjecture I: The sparse decomposition. SIAM JOURNAL ON DISCRETE MATHEMATICS, 31 (2). pp. 945-982. ISSN 0895-4801

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Abstract

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 k0 such that for every k > k0, 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 suitable combinatorial structure inside the decomposition, which we then use for embedding the tree. © 2017 the authors.

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