A bound on the inducibility of cycles

TL;DR

This paper proves an upper bound on the number of induced cycles of a fixed length in any n-vertex graph, confirming a conjecture with a new, tighter bound.

Contribution

It establishes a new upper bound of 2n^k/k^k on induced cycles, improving previous conjectures and advancing understanding of graph cycle structures.

Findings

Proved that every n-vertex graph has at most 2 n^k/k^k induced cycles of length k.

Confirmed a conjecture from 1975 with a tighter bound.

Provides a significant step in extremal graph theory related to induced cycles.

Abstract

In 1975, Pippenger and Golumbic conjectured that every n-vertex graph has at most $n^k/(k^k - k)$ induced cycles of length k for k at least 5. We prove that every n-vertex graph has at most $2 n^k/k^k$ induced cycles of length k.