Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth

TL;DR

This paper develops unified bijections for toroidal maps with prescribed face-degrees and essential girth, enabling enumeration and structural analysis of such maps through canonical orientations and decorated unicellular maps.

Contribution

It introduces a novel canonical orientation for toroidal d-angulations of essential girth d and derives bijections to decorated unicellular maps, extending planar bijections to the torus.

Findings

Explicit algebraic generating functions for rooted triangulations and quadrangulations on the torus.

Unified bijections generalizing planar cases to toroidal maps with face-degree and girth constraints.

Simplified bijections in the bipartite case as a parity specialization.

Abstract

We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d>=3 every toroidal d-angulation of essential girth d can be endowed with a certain "canonical" orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that, apart from the root-face contour, no other closed walk of length d encloses the root-face) and a family of decorated unicellular maps. The orientations and bijections can then be generalized, for any d>=1, to toroidal face-rooted maps of essential girth d with a root-face of degree d (and with the same root-face contour condition as for d-angulations), and they take a simpler form in the bipartite case, as a parity specialization. On the enumerative side we obtain explicit algebraic expressions for the generating functions of rooted essentially simple triangulations and bipartite quadrangulations on the torus. Our bijective constructions can be considered as toroidal counterparts of those obtained by Bernardi and the first author in the planar case, and they also build on ideas introduced by Despr\'e, Gon\c{c}alves and the second author for essentially simple triangulations, of imposing a balancedness condition on the orientations in genus 1.