 # Turán numbers of bipartite graphs plus an odd cycle

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

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Identification Number: 10.1016/j.jctb.2014.01.007

## 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  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 . 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 http://www.journals.elsevier.com/journal-of-combin... © 2014 Elsevier Inc. Mathematics Q Science > QA Mathematics Departments > Mathematics 28 Feb 2014 15:29 20 Nov 2019 11:03 http://eprints.lse.ac.uk/id/eprint/55928 View Item