### Abstract

We shall prove that if L is a 3‐chromatic (so called “forbidden”) graph, and —R^{n} is a random graph on n vertices, whose edges are chosen independently, with probability p, and —B^{n} is a bipartite subgraph of R^{n} of maximum size, —F^{n} is an L‐free subgraph of R^{n} of maximum size, then (in some sense) F^{n} and B^{n} are very near to each other: almost surely they have almost the same number of edges, and one can delete O_{p}(1) edges from F^{n} to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, F^{n} is almost surely bipartite.

Original language | English |
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Pages (from-to) | 599-622 |

Number of pages | 24 |

Journal | Journal of Graph Theory |

Volume | 14 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1990 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*14*(5), 599-622. https://doi.org/10.1002/jgt.3190140511

**Extremal subgraphs of random graphs.** / Babai, László; Simonovits, M.; Spencer, Joel.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 14, no. 5, pp. 599-622. https://doi.org/10.1002/jgt.3190140511

}

TY - JOUR

T1 - Extremal subgraphs of random graphs

AU - Babai, László

AU - Simonovits, M.

AU - Spencer, Joel

PY - 1990

Y1 - 1990

N2 - We shall prove that if L is a 3‐chromatic (so called “forbidden”) graph, and —Rn is a random graph on n vertices, whose edges are chosen independently, with probability p, and —Bn is a bipartite subgraph of Rn of maximum size, —Fn is an L‐free subgraph of Rn of maximum size, then (in some sense) Fn and Bn are very near to each other: almost surely they have almost the same number of edges, and one can delete Op(1) edges from Fn to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, Fn is almost surely bipartite.

AB - We shall prove that if L is a 3‐chromatic (so called “forbidden”) graph, and —Rn is a random graph on n vertices, whose edges are chosen independently, with probability p, and —Bn is a bipartite subgraph of Rn of maximum size, —Fn is an L‐free subgraph of Rn of maximum size, then (in some sense) Fn and Bn are very near to each other: almost surely they have almost the same number of edges, and one can delete Op(1) edges from Fn to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, Fn is almost surely bipartite.

UR - http://www.scopus.com/inward/record.url?scp=84986469590&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84986469590&partnerID=8YFLogxK

U2 - 10.1002/jgt.3190140511

DO - 10.1002/jgt.3190140511

M3 - Article

AN - SCOPUS:84986469590

VL - 14

SP - 599

EP - 622

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 5

ER -