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arXiv · 2601.04822

Asymptotic enumeration of constrained bipartite, directed and oriented graphs by degree sequence

Catherine Greenhill·Mahdieh Hasheminezhad·Isaiah Iliffe·Brendan D. McKay
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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.

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Catherine Greenhill, Mahdieh Hasheminezhad, Isaiah Iliffe, Brendan D. McKay. 2026-01-08. Asymptotic enumeration of constrained bipartite, directed and oriented graphs by degree sequence. https://arxiv.org/abs/2601.04822

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