  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 $q
\ge 2$, is maximized by a union of $K_{3,3}$'s. This proves the $d=3$ case of a
conjecture of Galvin and Tetali.
