Path correlations in a randomly oriented complete bipartite graph

Abstract

In a randomly oriented graph containing vertices $x$ and $y$, denote by $\{x\to y\}$ the event that there is a directed path from $x$ to $y$. We study the correlation between the events $\{x\to y\}$ and $\{y\to z\}$ for a (large) oriented complete bipartite graph with orientation chosen uniformly at random. We classify the cases of positive and negative correlation respectively in terms of the relative proportions of the sizes of the color classes of the graph.