# CSM

## Seminar

### A bijection for covered maps on orientable surfaces

Seminar Room 1, Newton Institute

#### Abstract

A map of genus g is a graph together with an embedding in the orientable surface of genus g. For instance, plane trees can be considered as maps of genus 0 and unicellular maps (maps with a single face) are a natural generalisation of plane trees to higher genus surfaces.

In this talk, we consider covered maps, which are maps together with a distinguished unicellular spanning submap. We will present a bijection between covered maps of genus g with n edges and pairs made of a plane tree with n edges and a bipartite unicellular map of genus g with n+1 edges. This bijection allows to recover bijectively some very elegant formulas by Mullin and by Lehman and Walsh. We will also show that our bijection generalises a bijection of Bouttier, Di Francesco and Guitter (which, in turns, generalises a previous bijection of Schaeffer) between bipartite maps and some classes of labelled trees.

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