On the inducibility of cycles

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.