Extremal results for odd cycles in sparse pseudorandom graphs

Abstract

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Г the generalized Turán density π _F (Г) denotes the relative density of a maximum subgraph of Г, which contains no copy of F. Extending classical Turán type results for odd cycles, we show that π _F (Г)=1/2 provided F is an odd cycle and Г is a sufficiently pseudorandom graph.

In particular, for (n,d,λ)-graphs Г, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval [−λ,λ], our result holds for odd cycles of length ℓ, provided

$$\lambda ^{\ell - 2} \ll \frac{{d^{\ell - 1} }} {n}\log (n)^{ - (\ell - 2)(\ell - 3)} .$$

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, Szabó, 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;λ)-graphs) shows that our assumption on Г is best possible up to the polylog-factor for every odd ℓ≥5.