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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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 (Other) http://arxiv.org/ © 2009 The authors Q Science > QA Mathematics Departments > Mathematics http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx 09 Apr 2010 14:20 http://eprints.lse.ac.uk/27676/