In mathematics, a dessin d'enfant (French for a "child's drawing", plural dessins d'enfants, "children's drawings") is a type of graph drawing used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.
Intuitively, a dessin d'enfant is simply a graph, with its vertices colored alternating black and white, embedded onto an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite. The faces of the embedding must be topological disks. The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.
Any dessin can provide the surface it is embedded on with a structure as a Riemann surface. It natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined over the field of algebraic numbers (when considered as algebraic curves). The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.
For a more detailed treatment of this subject, see Schneps (1994) or Lando & Zvonkin (2004).
Read more about Dessin D'enfant: Riemann Surfaces and Belyi Pairs, Maps and Hypermaps, Regular Maps and Triangle Groups, Trees and Shabat Polynomials, The Absolute Galois Group and Its Invariants