On the Generalized Turán Problem for Odd Cycles

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2416-2428, September 2024.
Abstract. In 1984, Erdős conjectured that the number of pentagons in any triangle-free graph on [math] vertices is at most [math], which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami et al. As an extension of this result for longer cycles, we prove that for each odd [math], the balanced blow-up of [math] (uniquely) maximizes the number of [math]-cycles among [math]-free graphs on [math] vertices, as long as [math] is sufficiently large. We also show that this is no longer true if [math] is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd [math], the balanced blow-up of [math] maximizes the number of [math]-cycles among graphs with a given number of vertices and no odd cycles of length less than [math]. We further show that if [math] and [math] are odd and [math] is sufficiently large compared to [math], then the balanced blow-up of [math] does not asymptotically maximize the number of [math]-cycles among [math]-free graphs on [math] vertices. This disproves a conjecture of Grzesik and Kielak.