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PreprintCC BY 4.0arXiv

Universal Asymptotics and Exact Enumeration of Eulerian Maps

  • Ahmad Barhoumi
  • Roozbeh Gharakhloo
  • Nathan Hayford
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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 three map characteristics, regardless of the choice of the degree sequence. The constant factor in this formula is related to the Painlevé I equation. Our methods combine for the first time 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≥ 1, non-regular (mixed-valence) maps.

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Cite

BibTeXRIS
BibTeX
@misc{barhoumi2026universal,
  title = {{Universal Asymptotics and Exact Enumeration of Eulerian Maps}},
  author = {Ahmad Barhoumi and Roozbeh Gharakhloo and Nathan Hayford},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.14342},
  eprint = {2607.14342},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.14342},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Universal Asymptotics and Exact Enumeration of Eulerian Maps
AU  - Ahmad Barhoumi
AU  - Roozbeh Gharakhloo
AU  - Nathan Hayford
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2607.14342
UR  - https://arxiv.org/abs/2607.14342
AB  - 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 three map characteristics, regardless of the choice of the degree sequence. The constant factor in this formula is related to the Painlevé I equation. Our methods combine for the first time 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≥ 1, non-regular (mixed-valence) maps.
ER  -