Graphs with Many Strong Orientations
Publication Date
6-21-2016
Document Type
Article
Organizational Units
Mathematics
Keywords
Sparse graph, High probability, Strong orientations, Strongly connected, Cheeger constant
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
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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.