Year: 2023
Type: preprint
Abstract: How many edges in an $n$-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 $n$-vertex graph with $2n^{3/2}$ edges contains such a cycle. We significantly improve this old bound by showing that $Ω(n\log^8n)$ 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.
Source: arXiv (Cornell University)
Institution: ETH Zurich
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G993971353
OpenAlex ID: W4380994353