Asymptotic enumeration of constrained bipartite, directed and oriented graphs by degree sequence
Combinatorics
2026-01-09 v1
Abstract
In the sufficiently sparse case, we find the probability that a uniformly random bipartite graph with given degree sequence contains no edge from a specified set of edges. This enables us to enumerate loop-free digraphs and oriented graphs with given in-degree and out-degree sequences, and obtain subgraph probabilities. Our theorems are not restricted to the near-regular case. As an application, we determine the expected permanent of sparse or very dense random matrices with given row and column sums; in the regular case, our formula holds over all densities. We also draw conclusions about the degrees of a random orientation of a random undirected graph with given degrees, including its number of Eulerian orientations.
Cite
@article{arxiv.2601.04822,
title = {Asymptotic enumeration of constrained bipartite, directed and oriented graphs by degree sequence},
author = {Catherine Greenhill and Mahdieh Hasheminezhad and Isaiah Iliffe and Brendan D. McKay},
journal= {arXiv preprint arXiv:2601.04822},
year = {2026}
}