#### Title

Graphs with Many Strong Orientations

#### Document Type

Article

#### Publication Date

6-21-2016

#### Keywords

Sparse graph, High probability, Strong orientations, Strongly connected, Cheeger constant

#### Organizational Units

College of Natual Science and Mathematics, Mathematics

#### 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.

#### Publication Statement

Copyright held by author or publisher. User is responsible for all copyright compliance.

#### Recommended Citation

Aksoy, Sinan, and Paul Horn. "Graphs with Many Strong Orientations." Siam Journal on Discrete Mathematics. 30.2 (2016): 1269-1282. Print. doi: 10.1137/15m1018885.