A bijection for covered maps on orientable surfaces

Unicellular maps are a natural generalisation of plane trees to higher genus surfaces. In this article we study covered maps, which are maps together with a distinguished unicellular spanning submap. We prove that the covered maps of genus g with n edges are in bijection with pairs made of a plane tree with n edges and a bipartite unicellular map of genus g with n +1 edges. This generalises to any genus the bijection given in [2] between planar tree-rooted maps (maps with a distinguished spanning tree) and pairs made of a tree with n edges and a tree with n + 1 edges. In the special case of genus 1, a duality argument allows us to obtain a bijective proof of a formula of Lehman and Walsh [4] about the number of tree-rooted maps of genus 1.