Universal Asymptotics and Exact Enumeration of Eulerian Maps

Ahmad Barhoumi; Roozbeh Gharakhloo; Nathan Hayford

Universal Asymptotics and Exact Enumeration of Eulerian Maps

Combinatorics 2026-07-15 v1 Mathematical Physics Complex Variables

Authors: Ahmad Barhoumi , Roozbeh Gharakhloo , Nathan Hayford

Abstract

We calculate the asymptotics of the number of connected, labeled, genus $g$ Eulerian maps with an arbitrary degree sequence, in the limit as the total number of vertices tends to infinity. This asymptotic is universal, in the sense that the leading order term depends on only finitely many map characteristics. The constant factor in this formula is related to the Painlev\'{e} I equation. Our methods combine the analysis of the recurrence coefficients associated to a particular family of orthogonal polynomials, and the theory of analytic combinatorics of several variables. We also derive an exact formula for the number of connected, labeled, genus $1$ Eulerian maps. These are the first results on this kind of enumeration problem for $g\geq 1$ , non-regular (mixed-valence) maps.

Cite

@article{arxiv.2607.14342,
  title  = {Universal Asymptotics and Exact Enumeration of Eulerian Maps},
  author = {Ahmad Barhoumi and Roozbeh Gharakhloo and Nathan Hayford},
  journal= {arXiv preprint arXiv:2607.14342},
  year   = {2026}
}