  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 $q\ge
5$. This proves the first case of a conjecture of the author, Perkins, Jenssen,
and Roberts on extensions of their methods, 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 $q \ge 5$ we prove that the
number of proper $q$-colourings of a $4$-regular graph is maximised by a union
of $K_{4,4}$'s.
