$C_5$ is almost a fractalizer

TL;DR

This paper determines the maximum number of induced 5-cycles in any n-vertex graph, showing that balanced blow-ups of a 5-cycle are extremal, with a special case for n=8 involving the M"obius ladder.

Contribution

It completes the exact extremal enumeration of induced 5-cycles, extending previous asymptotic results by applying and expanding flag algebra methods to small graphs.

Findings

Balanced blow-ups of a 5-cycle are extremal for most n.

The M"obius ladder is extremal for n=8.

The result confirms the conjecture for all n.

Abstract

We determine the maximum number of induced copies of a 5-cycle in a graph on $n$ vertices for every $n$. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for $n=8$, the M\"obius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices. This result completes work of Balogh, Hu, Lidick\'y, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.