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

Jan Hladký, János Komlós, Diana Piguet, Miklós Simonovits, Maya Stein, Endre Szemerédi

Research output: Contribution to journalArticle

10 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)945-982
Number of pages38
JournalSIAM Journal on Discrete Mathematics
Issue number2
Publication statusPublished - Jan 1 2017


  • Extremal graph theory
  • Graph decomposition
  • Loebl-Komlós-Sós conjecture
  • Regularity lemma
  • Sparse graph
  • Tree embedding

ASJC Scopus subject areas

  • Mathematics(all)

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