Counting Proper Colourings in 4-Regular Graphs via the Potts Model

摘要

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $qge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function.