### Abstract

This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any LεL. In this paper, we prove sharp results about the case L={O_{6}}, where O_{6} is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results. (a) The vertex set of almost every O_{6}-free graph can be partitioned into two classes of almost equal sizes, U1 and U2, where the graph spanned by U1 is a C4-free and that by U2 is P3-free. (b) Similar assertions hold when L is the family of all graphs with 6 vertices and 12 edges. (c) If H is a graph with a color-critical edge and Χ(H)=p+1, then almost every sH-free graph becomes p-chromatic after the deletion of some s-1 vertices, where sH is the graph formed by s vertex disjoint copies of H.These results are natural extensions of theorems of classical extremal graph theory. To show that results like those above do not hold in great generality, we provide examples for which the analogs of our results do not hold.

Original language | English |
---|---|

Pages (from-to) | 67-84 |

Number of pages | 18 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 101 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2011 |

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### Keywords

- Extremal graphs
- Graph counting
- Structure of H-free graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*101*(2), 67-84. https://doi.org/10.1016/j.jctb.2010.11.001

**The fine structure of octahedron-free graphs.** / Balogh, József; Bollobás, Béla; Simonovits, M.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 101, no. 2, pp. 67-84. https://doi.org/10.1016/j.jctb.2010.11.001

}

TY - JOUR

T1 - The fine structure of octahedron-free graphs

AU - Balogh, József

AU - Bollobás, Béla

AU - Simonovits, M.

PY - 2011/3

Y1 - 2011/3

N2 - This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any LεL. In this paper, we prove sharp results about the case L={O6}, where O6 is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results. (a) The vertex set of almost every O6-free graph can be partitioned into two classes of almost equal sizes, U1 and U2, where the graph spanned by U1 is a C4-free and that by U2 is P3-free. (b) Similar assertions hold when L is the family of all graphs with 6 vertices and 12 edges. (c) If H is a graph with a color-critical edge and Χ(H)=p+1, then almost every sH-free graph becomes p-chromatic after the deletion of some s-1 vertices, where sH is the graph formed by s vertex disjoint copies of H.These results are natural extensions of theorems of classical extremal graph theory. To show that results like those above do not hold in great generality, we provide examples for which the analogs of our results do not hold.

AB - This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any LεL. In this paper, we prove sharp results about the case L={O6}, where O6 is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results. (a) The vertex set of almost every O6-free graph can be partitioned into two classes of almost equal sizes, U1 and U2, where the graph spanned by U1 is a C4-free and that by U2 is P3-free. (b) Similar assertions hold when L is the family of all graphs with 6 vertices and 12 edges. (c) If H is a graph with a color-critical edge and Χ(H)=p+1, then almost every sH-free graph becomes p-chromatic after the deletion of some s-1 vertices, where sH is the graph formed by s vertex disjoint copies of H.These results are natural extensions of theorems of classical extremal graph theory. To show that results like those above do not hold in great generality, we provide examples for which the analogs of our results do not hold.

KW - Extremal graphs

KW - Graph counting

KW - Structure of H-free graphs

UR - http://www.scopus.com/inward/record.url?scp=78951485003&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78951485003&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2010.11.001

DO - 10.1016/j.jctb.2010.11.001

M3 - Article

AN - SCOPUS:78951485003

VL - 101

SP - 67

EP - 84

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -