Universal Asymptotics and Exact Enumeration of Eulerian Maps
arXiv:2607.14342
The paper derives universal asymptotic formulas for counting labeled Eulerian maps of any genus with arbitrary degree sequences, and provides an exact enumeration for genus‑1 maps, using orthogonal polynomial recurrences and multivariate analytic combinatorics.
Abstract
We calculate the asymptotics of the number of connected, labeled, genus 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 Painlevé 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 Eulerian maps. These are the first results on this kind of enumeration problem for , non-regular (mixed-valence) maps.