We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex degrees. The mapping from blossoming trees to maps is a generalization to unfixed genus of Schaeffer's closing construction for planar Eulerian maps. The inverse mapping relies on the existence of canonical orientations which allow to equip the maps with canonical spanning trees, as proved by Bernardi. Our bijection gives in particular (here in the Eulerian case) a combinatorial explanation to the striking similarity between the (infinite) recursive system of equations which determines the partition function of maps with unfixed genus (as obtained via matrix models and orthogonal polynomials) and that determining the partition function of planar maps. All the functions in the recursive system get a combinatorial interpretation as generating functions for maps endowed with particular multiple markings of their edges. This allows us in particular to give a combinatorial proof of some differential identities satisfied by these functions. We also consider face-colored Eulerian maps with unfixed genus and derive some striking identities between their generating functions and those of properly weighted marked maps. The same methodology is then applied to deal with m-regular bipartite maps with unfixed genus, leading to similar results. The case of cubic maps is also briefly discussed.

我们在不固定属的生根图家族和赋予其叶子任意正向匹配的所谓开花树木家族之间引入双射。我们首先关注具有受控顶点度的欧拉图。从开花的树到地图的映射是对Schaeffer平面Eulerian映射的闭合构造的不固定类的概括。逆映射依赖于规范方向的存在，这可以使地图配备规范的生成树，正如Bernardi所证明的那样。我们的双射尤其是（在欧拉情况下）对方程式（无限）递归系统之间惊人相似性的组合解释，该方程式确定了具有不固定属的映射的分区函数（通过矩阵模型和正交多项式获得）确定平面图的分区函数。递归系统中的所有功能都得到了组合的解释，即为具有特定边缘多个标记的地图生成功能。这尤其使我们能够给出这些函数满足的一些差分身份的组合证明。我们还考虑了具有不固定属的面色欧拉图，并在它们的生成函数与适当加权的标记图之间产生了一些惊人的标识。具有不固定属的m-正则二分图，导致相似的结果。还简要讨论了三次映射的情况。