GRAPHS WITH MANY STRONG ORIENTATIONS

作者：Aksoy Sinan^*; Horn Paul

来源：SIAM Journal on Discrete Mathematics, 2016, 30(2): 1269-1282.

DOI：10.1137/15M1018885

• strong orientations
• strongly connected
• Cheeger constant

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) 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.

• 出版日期2016