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THE TYPICAL STRUCTURE OF MAXIMAL TRIANGLE-FREE GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  09 October 2015

JÓZSEF BALOGH ,
HONG LIU ,
ŠÁRKA PETŘÍČKOVÁ  and
MARYAM SHARIFZADEH
JÓZSEF BALOGH
Affiliation:
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; jobal@math.uiuc.edu, hliu36@illinois.edu, petrckv2@illinois.edu, sharifz2@illinois.edu
HONG LIU
Affiliation:
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; jobal@math.uiuc.edu, hliu36@illinois.edu, petrckv2@illinois.edu, sharifz2@illinois.edu
ŠÁRKA PETŘÍČKOVÁ
Affiliation:
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; jobal@math.uiuc.edu, hliu36@illinois.edu, petrckv2@illinois.edu, sharifz2@illinois.edu
MARYAM SHARIFZADEH
Affiliation:
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; jobal@math.uiuc.edu, hliu36@illinois.edu, petrckv2@illinois.edu, sharifz2@illinois.edu

Abstract

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Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most $2^{n^{2}/8+o(n^{2})}$$n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.

Our proof uses the Ruzsa–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_{3}$s, which is of independent interest.

MSC classification

Primary: 05C35: Extremal problems 05C30: Enumeration in graph theory 05C69: Dominating sets, independent sets, cliques
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e20
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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