Inducibility of cycles of length at least six

Determine the inducibility I(C_m) for every cycle C_m with m≥6, including whether the iterated blow-up construction conjectured by Pippenger and Golumbic is extremal.

Background

The paper describes the inducibility problem for cycles as a major unresolved direction. Pippenger and Golumbic conjectured an iterated blow-up construction for cycles of length at least five; the case m=5 has been proved, while the cases m≥6 remain unresolved.

References

Another outstanding problem is to determine $I(C_m)$ for $m \geq 6$. Pippenger and Golumbic conjectured that an `iterated blow-up' of $C_m$ is extremal for $m \geq 5$, which would yield $I(C_m)=m!/(mm-m)$:

— The semi-inducibility problem  (2501.09842 - Basit et al., 16 Jan 2025) in Section 2.2, “The inducibility problem”