The approximate Loebl-Komlós-Sós conjecture IV: Embedding techniques and the proof of the main result

Jan Hladký, János Komlós, Diana Piguet, M. Simonovits, Maya Stein, E. Szemerédi

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

This is the last of a series of four papers in which 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. In the first two papers of this series, we decomposed the host graph G and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlós-Sós conjecture contains one of ten specific configurations. In this paper we embed the tree T in each of the ten configurations.

Original languageEnglish
Pages (from-to)1072-1148
Number of pages77
JournalSIAM Journal on Discrete Mathematics
Volume31
Issue number2
DOIs
Publication statusPublished - Jan 1 2017

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'The approximate Loebl-Komlós-Sós conjecture IV: Embedding techniques and the proof of the main result'. Together they form a unique fingerprint.

  • Cite this