# On the inducibility of cycles

## TL;DR

This paper improves the upper bound on the maximum number of induced cycles of length k in an n-vertex graph, advancing towards a longstanding conjecture about the tightness of a known lower bound.

## Contribution

It establishes a new upper bound of (128e/81)·(n/k)^k for the number of induced k-cycles, the first progress since the conjecture was posed.

## Key findings

- New upper bound tighter than previous results
- Progress towards confirming the conjecture of tightness
- First improvement since the conjecture was introduced

## Abstract

In 1975 Pippenger and Golumbic proved that any graph on $n$ vertices admits at most $2e(n/k)^k$ induced $k$-cycles. This bound is larger by a multiplicative factor of $2e$ than the simple lower bound obtained by a blow-up construction. Pippenger and Golumbic conjectured that the latter lower bound is essentially tight. In the present paper we establish a better upper bound of $(128e/81) \cdot (n/k)^k$. This constitutes the first progress towards proving the aforementioned conjecture since it was posed.