Abstract    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 [O. Bernardi. Bijective counting of tree-rooted maps and shuffles of parenthesis systems.  Electron. J. Combin. , 14(1): Research Paper 9, 36 pp., 2007] 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.
['Olivier Bernardi', 'Guillaume Chapuy']
A bijection for covered maps on orientable surfaces