Directed graphs without short cycles
Jacob Fox, Peter Keevash, Benny Sudakov

TL;DR
This paper establishes bounds on feedback arc sets in directed graphs with large girth, extending previous results and answering questions about cycle lengths in such graphs.
Contribution
It proves a tight bound on feedback arc sets in high girth digraphs and explores cycle length properties related to feedback arc set size.
Findings
Bound on feedback arc set size in high girth digraphs.
Existence of cycles of almost any length in graphs with large feedback arc sets.
Structural characterization of graphs with large feedback arc sets.
Abstract
For a directed graph without loops or parallel edges, let denote the size of the smallest feedback arc set, i.e., the smallest subset such that has no directed cycles. Let be the number of unordered pairs of vertices of which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least satisfies , where is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed and sufficiently large , if is a digraph with vertices and , then for any it contains a directed cycle…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
