Georgia Institute of Technology College of Sciences

A fundamental question in topological graph theory is as follows: Given a surface S and an integer t > 0, which graphs drawn in S are t-colorable? We say that a graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In 1993, Carsten Thomassen showed that there are only finitely many six-critical graphs on a fixed surface with Euler genus g. In this talk, I will describe a new short proof of this fact. In addition, I will describe some structural lemmas that were useful to the proof and describe a list-coloring extension that is helpful to ongoing work that there are finitely many six-list-critical graphs on a fixed surface. This is a joint project with Ken-ichi Kawarabayashi of the National Institute of Informatics, Tokyo.