Dessins d’enfants with a given bipartite graph

Abstract

Let G \mathcal {G} be a finite connected bipartite graph. We describe the set D e s s G { \mathcal Dess}_{\mathcal G} consisting of all the isomorphism classes of the dessins d’enfants whose underlying bipartite graph is isomorphic to G \mathcal {G} . Such a description is explicit and it permits, for each dessin, to provide some of its Galois invariants, such as its passport, its automorphism group, its monodromy group and its Z-orientability property.