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Extremal subgraphs of random graphs: an extended version

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

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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)
Official URL:
Additional Information: © 2009 The authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 09 Apr 2010 14:20
Last Modified: 15 Sep 2023 22:10

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