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Triangle-Free Subgraphs of Random Graphs

Part of: Graph theory

Published online by Cambridge University Press:  14 August 2017

PETER ALLEN ,
JULIA BÖTTCHER ,
YOSHIHARU KOHAYAKAWA  and
BARNABY ROBERTS
PETER ALLEN
Affiliation:
Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: p.d.allen@lse.ac.uk, j.boettcher@lse.ac.uk, b.j.roberts@lse.ac.uk)
JULIA BÖTTCHER
Affiliation:
Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: p.d.allen@lse.ac.uk, j.boettcher@lse.ac.uk, b.j.roberts@lse.ac.uk)
YOSHIHARU KOHAYAKAWA
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508–090 São Paulo, Brazil (e-mail: yoshi@ime.usp.br)
BARNABY ROBERTS
Affiliation:
Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: p.d.allen@lse.ac.uk, j.boettcher@lse.ac.uk, b.j.roberts@lse.ac.uk)
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Abstract

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p−1n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p−1n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 141 - 161
Copyright
Copyright © Cambridge University Press 2017 

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