Bipartite graph embeddings, Riemann surfaces and Galois groups

Bipartite graphs can model matching problems, hypergraphs and relations between sets. Studying their surface embeddings is an old tradition, illustrated by the utilities problem. I shall show how such embeddings can be described by pairs of permutations, and how this leads to a classification of the regular (most symmetric) embeddings of complete bipartite graphs. This is joint work with Du, Kwak, Nedela and Škoviera, building on earlier group-theoretic results of Hall, Huppert and Wielandt. Surface embeddings of bipartite graphs (called dessins d’enfants by Grothendieck) also give a link between compact Riemann surfaces and algebraic number fields, providing a faithful representation of the Galois group of the field of algebraic numbers, an important profinite group. I shall describe joint work with Streit and Wolfart on how this group acts on regular embeddings of complete bipartite graphs.