Brightwell, Graham, Panagiotou, Konstantinos and Steger, Angelika
(2009)
*Extremal subgraphs of random graphs: an extended version.*
.
arXiv.

## Abstract

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question of Babai, Simonovits and Spencer (Journal of Graph Theory, 1990). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with $M$ edges, where $M >> n$, is ``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M}$, we can obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices between the parts of $C$.

Item Type: | Monograph (Report) |
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Official URL: | http://arxiv.org/ |

Additional Information: | © 2009 The authors |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 09 Apr 2010 14:20 |

Last Modified: | 04 Sep 2021 23:06 |

URI: | http://eprints.lse.ac.uk/id/eprint/27676 |

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