### Abstract

Let L be a finite family of graphs. We describe the typical structure of L-free graphs, improving our earlier results (Balogh et al., J Combinat Theory Ser B 91 (2004), 1-24) on the Erdo{double acute}s- Frankl-Rödl theorem (Erdo{double acute}s et al., Graphs Combinat 2 (1986), 113-121), by proving our earlier conjecture that, for p = p(L) = min _{L∈L} X(L) - 1, the structure of almost all L-free graphs is very similar to that of a random subgraph of the Turán graph T _{n,p}. The "similarity" is measured in terms of graph theoretical parameters of L.

Original language | English |
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Pages (from-to) | 305-318 |

Number of pages | 14 |

Journal | Random Structures and Algorithms |

Volume | 34 |

Issue number | 3 |

DOIs | |

Publication status | Published - máj. 2009 |

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### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*34*(3), 305-318. https://doi.org/10.1002/rsa.20242

**The typical structure of graphs without given excluded subgraphs.** / Balogh, József; Bollobás, Béla; Simonovits, M.

Research output: Article

*Random Structures and Algorithms*, vol. 34, no. 3, pp. 305-318. https://doi.org/10.1002/rsa.20242

}

TY - JOUR

T1 - The typical structure of graphs without given excluded subgraphs

AU - Balogh, József

AU - Bollobás, Béla

AU - Simonovits, M.

PY - 2009/5

Y1 - 2009/5

N2 - Let L be a finite family of graphs. We describe the typical structure of L-free graphs, improving our earlier results (Balogh et al., J Combinat Theory Ser B 91 (2004), 1-24) on the Erdo{double acute}s- Frankl-Rödl theorem (Erdo{double acute}s et al., Graphs Combinat 2 (1986), 113-121), by proving our earlier conjecture that, for p = p(L) = min L∈L X(L) - 1, the structure of almost all L-free graphs is very similar to that of a random subgraph of the Turán graph T n,p. The "similarity" is measured in terms of graph theoretical parameters of L.

AB - Let L be a finite family of graphs. We describe the typical structure of L-free graphs, improving our earlier results (Balogh et al., J Combinat Theory Ser B 91 (2004), 1-24) on the Erdo{double acute}s- Frankl-Rödl theorem (Erdo{double acute}s et al., Graphs Combinat 2 (1986), 113-121), by proving our earlier conjecture that, for p = p(L) = min L∈L X(L) - 1, the structure of almost all L-free graphs is very similar to that of a random subgraph of the Turán graph T n,p. The "similarity" is measured in terms of graph theoretical parameters of L.

KW - Extremal graphs

KW - Graph counting

KW - Structure of H-free graphs

UR - http://www.scopus.com/inward/record.url?scp=67649112244&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649112244&partnerID=8YFLogxK

U2 - 10.1002/rsa.20242

DO - 10.1002/rsa.20242

M3 - Article

VL - 34

SP - 305

EP - 318

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -