Graphs with many strong orientations
Sinan Aksoy, Paul Horn

TL;DR
This paper demonstrates that under mild degree and Cheeger constant conditions, a sparse graph can be randomly oriented to produce a strongly connected directed graph with high probability, extending understanding of graph orientations.
Contribution
It establishes new conditions involving minimum degree and Cheeger constant that ensure many strong orientations in sparse graphs, including tightness results.
Findings
Random orientations produce strongly connected graphs under given conditions.
Cheeger constant bounds can be replaced by spectral conditions.
Minimum degree condition is shown to be tight.
Abstract
We establish mild conditions under which a possibly irregular, sparse graph has "many" strong orientations. Given a graph on vertices, orient each edge in either direction with probability independently. We show that if satisfies a minimum degree condition of and has Cheeger constant at least , then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we provide an explicit construction to show our minimum degree condition is tight while the Cheeger constant bound is tight up to a factor.
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.
