Blossoming bijection for higher-genus maps

TL;DR

This paper generalizes Schaeffer's bijection from planar maps to higher-genus surfaces, providing a canonical orientation and a new bijective proof of a rationality result for map generating series.

Contribution

It introduces a bijection between higher-genus bicolorable maps and unicellular maps, extending Schaeffer's planar map bijection to arbitrary genus surfaces.

Findings

Established a canonical orientation for higher-genus maps

Developed a local opening algorithm for these maps

Provided a bijective proof of the rationality of higher-genus map generating series

Abstract

In 1997, Schaeffer described a bijection between Eulerian planar maps and some trees. In this work we generalize his work to a bijection between bicolorable maps on a surface of any fixed genus and some unicellular maps with the same genus. An important step of this construction is to exhibit a canonical orientation for maps, that allows to apply the same local opening algorithm as Schaeffer. As an important byproduct, we obtain the first bijective proof of a result of Bender and Canfield from 1991, when they proved that the generating series of maps in higher genus is a rational function of the generating series of planar maps.