Bipartite graphs and their dessins d'enfants

TL;DR

This paper presents an algorithm to compute all dessins d'enfants associated with finite connected bipartite graphs, including their automorphism groups, monodromy groups, and duality types, linking graph theory with algebraic geometry.

Contribution

It introduces a novel algorithm for systematically computing dessins d'enfants from bipartite graphs and analyzing their symmetry and duality properties.

Findings

Algorithm successfully computes all dessins d'enfants for given bipartite graphs.

Provides detailed automorphism and monodromy group data for each dessin.

Classifies dessins based on duality types.

Abstract

Each finite and connected bipartite graph induces a finite collection of non-isomorphic dessins d'enfants, that is, $2$-cell embeddings of it into some closed orientable surface. We describe an algorithm to compute all these dessins d'enfants, together their automorphims group, monodromy group and duality type.