On the maximum number of odd cycles in graphs without smaller odd cycles

TL;DR

This paper establishes an upper bound on the number of specific odd cycles in graphs lacking smaller odd cycles, generalizing earlier results and providing a non-computer-assisted proof for the maximum cycle count.

Contribution

It generalizes the maximum cycle count bounds for graphs without small odd cycles, extending Erdős's conjecture and asymptotically determining related Turán numbers.

Findings

Proves an upper bound of (n/k)^k cycles of length k in such graphs

Generalizes previous results on pentagons in triangle-free graphs

Provides a non-computer-assisted proof for the bounds

Abstract

We prove that for each odd integer $k \geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles of length $k$. This generalizes the previous results on the maximum number of pentagons in triangle-free graphs, conjectured by Erd\H{o}s in 1984, and asymptotically determines the generalized Tur\'an number $\mathrm{ex}(n,C_k,C_{k-2})$ for odd $k$. In contrary to the previous results on the pentagon case, our proof is not computer-assisted.