Tutte 100th Colloquium - Carsten Thomassen

Title: Graph colorings, flows and decompositions

Abstract:

We discuss problems and results that are related to the Tutte polynomial, such as Tutte's flow conjectures, the chromatic polynomial, and the Merino-Welsh conjecture. We show that modulo k analogue of Tutte's f-factor theorem also fits in this framework. We consider the following question: For which real numbers t is the chromatic polynomial of every planar graph positive at t? Birkhoff and Lewis proved that every number greater than or equal to 5 has that property. The 4-color theorem says that 4 has that property. A celebrated result of Tutte says that 3.618... also has that property. Are there others?