On extremal subgraphs of random graphs

Authors:
Graham Brightwell;Konstantinos Panagiotou;Angelika Steger
Affiliations:
London School of Economics, London, United Kingdom;Institute of Theoretical Computer Science, ETH Zurich, Zurich, Switzerland and SNF;Institute of Theoretical Computer Science, ETH Zurich, Zurich, Switzerland
Venue:
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2007

Citing 4
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Extremal subgraphs of random graphs

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Abstract

Let κl denote the complete graph on l vertices. We prove that there is a constant c = c(l) 0, such that whenever p ≥ n-c, with probability tending to 1 when n goes to infinity, every maximum κl-free subgraph of the binomial random graph Gn, p is (l-1)-partite. This answers a question of Babai, Simonovits and Spencer [3]. 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 Gn, M, we can obtain all maximum cuts by moving at most O (√n3/M) vertices between the parts of C.