Counting proper colourings in 4-regular graphs via the Potts model

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 q >= 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. The zero-temperature limit gives upper and lower bounds on the number of proper q-colourings of 4-regular graphs, which almost proves the case d = 4 of a conjecture of Galvin and Tetali. For any q >= 5 we prove that the number of proper q-colourings of a 4-regular graph is maximised by a union of K-4,K-4's.