## Random embeddings of graphs

### Abstract

It is a classical problem to determine the minimum and maximum genus of a $2$-cell embedding of a graph $G$. However there will be many other embeddings with a genus between these two values: we will study the average genus across all of these embeddings.

We give an overview of recent work on the average genus for fixed graphs. We also discuss extensions of the problem to random graphs and random maps, and links with representation theory.