Affiliation:
1. Department of Mathematics ETH Zürich Switzerland
2. Mathematical Institute University of Oxford Oxford United Kingdom
Abstract
AbstractHow 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.
Funder
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Cited by
2 articles.
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1. Sublinear Expanders and Their Applications;Surveys in Combinatorics 2024;2024-06-13
2. The Extremal Number of Cycles with All Diagonals;International Mathematics Research Notices;2024-04-18