  A multi-component electron model on a lattice is constructed whose ground
state exhibits a spontaneous ordering which follows the rule of map-coloring
used in the solution of the four color problem. The number of components is
determined by the Euler characteristics of a certain surface into which the
lattice is embedded. Combining the concept of chromatic polynomials with the
Heawood-Ringel-Youngs theorem, we derive an index theorem relating the
degeneracy of the ground state with a hidden topology of the lattice. The
system exhibits coloring transition and hidden-topological structure
transition. The coloring phase exhibits a topological order.
