A bijection for covered maps on orientable surfaces

Olivier Bernardi (1) , Guillaume Chapuy (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

A covered map of genus g is a map of genus g with a spanning unicellular submap (by unicellular, we mean that the submap has only one border, but it can have genus lower than g). We compute the number of covered maps of genus g with n edges thanks to a bijection, that generalizes Bernardi's plane construction (EJC, Vol 14, 2007). From the special case of genus 1, and a duality argument, we obtain a bijective proof of a formula of Lehman and Walsh for the number of toroidal tree-rooted maps (maps with a spanning tree).

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Guillaume Chapuy. A bijection for covered maps on orientable surfaces. The International Conference on Topological and Geometric Graph Theory, 2008, France. pp.63--68, ⟨10.1016/j.endm.2008.06.010⟩. ⟨hal-00713488⟩