I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck's dessins d´enfants. My background is in maps (graph embeddings), and I got fascinated about their application to this field.

As an example, I found it completely stunning when I saw the first time that the least genus embedding of $K_{n,n,n}$ (the complete tripartite graph with $3n$ vertices), when viewed as a dessin d'enfants, specifies the Fermat curve. (cf. Jones & Singermann "Maps, Hypermaps, and Triangle Groups" in: Schneps (ed.): The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Notes Series 200, Cambridge University Press, Cambridge 1994)

So I would like to look for more areas of potential application of graph embeddings to dessins d'enfants, but don't have a good overview yet over recent developments in this field.

Would you be able to recommend a recent survey of dessins d'enfants, or a good recent text to me?

I am not sure whether I am using the right tags for this, please feel free to adjust!

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph embeddings), and I got fascinated about their application to dessins d´enfants.

As an example, I found it completely stunning when I saw the first time that the least genus embedding of $K_{n,n,n}$ (the complete tripartite graph with $3n$ vertices), when viewed as a dessin d'enfants, specifies the Fermat curve. (cf. Jones & Singermann "Maps, Hypermaps, and Triangle Groups" in: Schneps (ed.): The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Notes Series 200, Cambridge University Press, Cambridge 1994)

So I would like to look for more areas of potential application of graph embeddings to dessins d'enfants, but don't have a good overview yet over recent developments in this field.

Would you be able to recommend a recent survey of dessins d'enfants, or a good recent text to me?

I am not sure whether I am using the right tags for this, please feel free to adjust!