A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

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Abstract

Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turán's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)-t edges contains a p-partite subgraph with at least e(Tn,p)-2t edges. As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemerédi's regularity lemma) for the classical stability result of Simonovits [25].

Original languageEnglish
Pages (from-to)66-71
Number of pages6
JournalJournal of Combinatorial Theory. Series B
Volume115
DOIs
Publication statusPublished - Nov 1 2015

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Extremal Graphs
Corollary
Regularity
Regularity Lemma
Graph in graph theory
Subgraph
Lemma
Denote
Theorem

Keywords

  • Extremal graphs
  • Stability
  • Turán number

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

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title = "A proof of the stability of extremal graphs, Simonovits' stability from Szemer{\'e}di's regularity",
abstract = "Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Tur{\'a}n's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)-t edges contains a p-partite subgraph with at least e(Tn,p)-2t edges. As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemer{\'e}di's regularity lemma) for the classical stability result of Simonovits [25].",
keywords = "Extremal graphs, Stability, Tur{\'a}n number",
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language = "English",
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T1 - A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

AU - Füredi, Z.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turán's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)-t edges contains a p-partite subgraph with at least e(Tn,p)-2t edges. As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemerédi's regularity lemma) for the classical stability result of Simonovits [25].

AB - Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turán's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)-t edges contains a p-partite subgraph with at least e(Tn,p)-2t edges. As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemerédi's regularity lemma) for the classical stability result of Simonovits [25].

KW - Extremal graphs

KW - Stability

KW - Turán number

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