Definition and first examples

Definition 4.1 A dessin d'enfant, or simply a dessin, is a pair (X, D) where X is an oriented compact topological surface, and D ⊂ X is a finite graph such that:

1. (i) D is connected.

2. (ii) D is bicoloured, i.e. the vertices have been given either white or black colour and vertices connected by an edge have different colours.

3. (iii) X \ D is the union of finitely many topological discs, which we call faces of D.

The genus of (X, D) is simply the genus of the topological surface X.

We consider two dessins (X1, D1) and (X2, D2) equivalent when there exists an orientation-preserving homeomorphism from X1 to X2 whose restriction to D1 induces an isomorphism between the coloured graphs D1 and D2.

Remark 4.2 In fact condition (i) is a consequence of condition (iii) as it is fairly obvious that any path in X connecting two given points of D is homotopic to a path supported on the boundary of the faces encountered along the way.

Remark 4.3 Some authors remove condition (ii) with the understanding that to any (single-coloured) graph satisfying conditions (i) and (iii), a dessin is associated by placing a new vertex in the middle of each edge. This process produces only dessins where all the white vertices have degree 2, a restriction that looks rather unnatural from the point of view of bicoloured graphs. These graphs are classically known as maps (see [JS78] and the references given there), and the associated dessins are the ones originally introduced by Grothendieck [Gro97].