Graphs with many strong orientations

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 $G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\log_2{n}$ and has Cheeger constant at least $c_2\frac{\log_2\log_2{n}}{\log_2{n}}$, 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 $\log_2\log_2{n}$ factor.