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 [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.