Extremes of the internal energy of the Potts model on cubic graphs

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q\ge 2$. 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 $q \ge 2$,ismaximizedbyaunionof$K_{3,3}${}^{\mathrm{\prime }}s\mathrm{.}Thisprovesthe$d=3$ case of a conjecture of Galvin and Tetali.