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 $d\ge 3$ 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 $d\ge 1$, 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 Despré, Gonçalves and the second author for essentially simple triangulations, of imposing a balancedness condition on the orientations in genus 1.

我们提出了圆环上的地图的统一双射，并控制了面部的度数和基本周长（周期性平面表示的周长）。第一步是证明$d\ge 3$每个环线d的基本周长d角都可以赋予一定的“规范”方向（以半边的权重形式表示）。使用结构由Bernardi的和Chapuy的适应，我们就可以导出然后之间的双射面根环形d必不可少周长-angulations d（其条件是，除了根脸部轮廓，长度的任何其他封闭的步行d包含根面）和一系列修饰的单细胞图。然后可以将方向和双射概括为$d\ge \mathrm{1个}$到必要的周长环形面层次映射d与程度的根面d（与作为相同的根面轮廓条件d -angulations），和他们在采取二分情况下的简单的形式，作为一个奇偶专业化。在枚举方面，我们获得了显式的代数表达式，用于表示环上生根的本质上简单的三角剖分和二分方三角剖分的生成函数。我们的双射构造可被视为与Bernardi和第一作者在平面情况下获得的双射构造相对应，并且它们也基于Despré，Gonçalves和第二作者针对基本简单的三角剖分引入的思想，即将平衡条件强加于平面。属的方向1。