Extremal results for odd cycles in sparse pseudorandom graphs

Author(s)

Aigner-Horev, Elad ^[1] ; Han, Hiep ^[2] ; Schacht, Mathias ^[1]

Affiliation

^[1] Univ Hamburg, Fachbereich Math, D-20146 Hamburg - Germany

^[2] Univ Sao Paulo, Inst Matemat & Estat, BR-05508090 Sao Paulo - Brazil

• Document type: Journal article
• Source: COMBINATORICA; v. 34, n. 4, p. 379-406, AUG 2014.
• Web of Science Citations: 0

Abstract

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Gamma the generalized Turan density pi(F) (Gamma) denotes the relative density of a maximum subgraph of Gamma, which contains no copy of F. Extending classical Turan type results for odd cycles, we show that pi(F) (Gamma)=1/2 provided F is an odd cycle and Gamma is a sufficiently pseudorandom graph. In particular, for (n,d,lambda)-graphs Gamma, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval {[}-lambda,lambda], our result holds for odd cycles of length l, provided lambda(l-2)<< dl-1/nlog(n)(-(l-2)(l-3)). Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szab, and Vu, who addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free (n,d;lambda)-graphs) shows that our assumption on Gamma is best possible up to the polylog-factor for every odd l >= 5. (AU)

• FAPESP's process: 10/16526-3 - Quasi-random hypergraphs and spanning subhypergraph containment
• Grantee: Hiep Han
• Support Opportunities: Scholarships in Brazil - Post-Doctoral