Allen, Peter, Keevash, Peter, Sudakov, Benny and Verstraëte, Jacques
(2014)
*Turán numbers of bipartite graphs plus an odd cycle.*
Journal of Combinatorial Theory, Series B, 106.
pp. 134-162.
ISSN 0095-8956

## Abstract

For an odd integer k, let Ck={C3,C5,…,Ck}Ck={C3,C5,…,Ck} denote the family of all odd cycles of length at most k and let CC denote the family of all odd cycles. Erdős and Simonovits [10] conjectured that for every family FF of bipartite graphs, there exists k such that ex(n,F∪Ck)∼ex(n,F∪C)ex(n,F∪Ck)∼ex(n,F∪C) as n→∞n→∞. This conjecture was proved by Erdős and Simonovits when F={C4}F={C4}, and for certain families of even cycles in [14]. In this paper, we give a general approach to the conjecture using Scott's sparse regularity lemma. Our approach proves the conjecture for complete bipartite graphs K2,tK2,t and K3,3K3,3: we obtain more strongly that for any odd k⩾5k⩾5, ex(n,F∪{Ck})∼ex(n,F∪C)ex(n,F∪{Ck})∼ex(n,F∪C) and we show further that the extremal graphs can be made bipartite by deleting very few edges. In contrast, this formula does not extend to triangles – the case k=3k=3 – and we give an algebraic construction for odd t⩾3t⩾3 of K2,tK2,t-free C3C3-free graphs with substantially more edges than an extremal K2,tK2,t-free bipartite graph on n vertices. Our general approach to the Erdős–Simonovits conjecture is effective based on some reasonable assumptions on the maximum number of edges in an m by n bipartite FF-free graph.

Item Type: | Article |
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Official URL: | http://www.journals.elsevier.com/journal-of-combin... |

Additional Information: | © 2014 Elsevier Inc. |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 28 Feb 2014 15:29 |

Last Modified: | 20 Mar 2019 02:22 |

URI: | http://eprints.lse.ac.uk/id/eprint/55928 |

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