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Dessins d'enfants can be regarded as bipartite graphs embed- ded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve dened over a num- ber eld). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge{transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the elds of denition for all curves with regular dessins of genus 2 to 18.

Galois actions on regular dessins of small genera