Cycles with many chords

Abstract

How many edges in an -vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any -vertex graph with edges contains such a cycle. We significantly improve this old bound by showing that edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.