Coloring the Voronoi tessellation of lattices

In this paper we define the chromatic number of a lattice: It is least colors one needs to color interiors cells Voronoi tessellation lattice so that no two sharing facet are same color. We compute root lattices, their duals, and Leech lattice, consider lattices Voronoi's first kind, investigate asymptotic behaviour when dimension tends infinity. introduce spectral lower bound for in spirit Hoffman's finite graphs. relate it character theory corresponding Lie groups.