number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic
graphs at every temperature and for all. This immediately implies corresponding tight
bounds on the anti‐ferromagnetic Potts partition function. Taking the zero‐temperature limit
gives new results in extremal combinatorics: the number of q‐colorings of a 3‐regular graph,
for any, is maximized by a union of's. This proves the d= 3 case of a conjecture of Galvin and …