Graph Colouring: A Modern Perspective

The Problem of colouring graphs goes back to De Morgan, who also conjectured the famous Four Colour Theorem, proved by Appel and Haken in 1976 with the aid of a computer. However, the theorem has so far defied any theoretical attempt to prove it. Interpreting the four-colour theorem as the colouring of trivalent dual graphs, Penrose suggested an alternative point of view that was developed later by Baldridge and McCarty in the language of categorification and topological quantum field theory. However, from the speaker’s point of view, a significant step was taken by Kronheimer and Mrowka, who used gauge theory to reformulate the idea of Tait-Colouring. The Conjecture made by them led Khovanov, Robert, and Wagner to pursue the colouring problem using foam evaluation. This later work and the work of Baldridge inspired the author to look at it from another very different perspective of topological defects, the subject that is tied with the concept of generalised symmetry in QFT (Runkel, Del-Zetto, etc). This talk is going to be a review of these approaches, and I intend to keep it colloquium style.