  The conjecture is verified that the optimum, energy minimizing magnetic flux
for a half-filled band of electrons hopping on a planar, bipartite graph is
$\pi$ per square plaquette. We require {\it only} that the graph has
periodicity in one direction and the result includes the hexagonal lattice
(with flux 0 per hexagon) as a special case. The theorem goes beyond previous
conjectures in several ways: (1) It does not assume, a-priori, that all
plaquettes have the same flux (as in Hofstadter's model); (2) A Hubbard type
on-site interaction of any sign, as well as certain longer range interactions,
can be included; (3) The conclusion holds for positive temperature as well as
the ground state; (4) The results hold in $D \geq 2$ dimensions if there is
periodicity in $D-1$ directions (e.g., the cubic lattice has the lowest energy
if there is flux $\pi$ in each square face).
