The distribution of the length of the longest path in random acyclic orientations of a complete bipartite graph

TL;DR

This paper analyzes the distribution of the longest path length in random acyclic orientations of complete bipartite graphs, providing asymptotic Gaussian behavior and precise mean and variance calculations.

Contribution

It introduces a probability generating function for the longest path length and applies analytic combinatorics to derive asymptotic results for equal part sizes.

Findings

Distribution is asymptotically Gaussian

Derived precise asymptotics for mean and variance

Addresses a question posed by Peter J. Cameron

Abstract

Randomly sampling an acyclic orientation on the complete bipartite graph $K_{n,k}$ with parts of size $n$ and $k$, we investigate the length of the longest path. We provide a probability generating function for the distribution of the longest path length, and we use Analytic Combinatorics to perform asymptotic analysis of the probability distribution in the case of equal part sizes $n = k$ tending toward infinity. We show that the distribution is asymptotically Gaussian, and we obtain precise asymptotics for the mean and variance. These results address a question asked by Peter J. Cameron. Keywords: bipartite graph, directed graph, random graph, acyclic orientation, poly-Bernoulli numbers, lonesum matrices, generating function, analytic combinatorics, asymptotics.