Generalized Tur\'an problem for directed cycles

TL;DR

This paper investigates the maximum number of directed cycles of a fixed length in large oriented graphs that avoid longer directed cycles, establishing asymptotic bounds and exact values for specific cases.

Contribution

It determines the order of magnitude of the generalized Turán problem for directed cycles and provides exact results for certain pairs of cycle lengths.

Findings

Established the order of magnitude for all pairs of cycle lengths.

Calculated exact extremal values for specific pairs of cycle lengths.

Showed the diversity of extremal constructions for small cycle lengths.

Abstract

For integers $k, \ell \geq 3$, let $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\ell$. We establish the order of magnitude of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for every $k$ and $\ell$ and determine its value up to a lower error term when $k \nmid \ell$ and $\ell$ is large enough. Additionally, we calculate the value of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for some other specific pairs $(k, \ell)$ showing that a diverse class of extremal constructions can appear for small values of $\ell$.