Periodic planar disc packings

Several conditions are given when a packing of equal discs in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any collectively jammed packing, whose graph does not consist of all triangles, and the torus lattice is the standard triangular lattice, is at most (n/(n + 1))π√12, where n is the number of packing discs in the torus. Several classes of collectively jammed packings are presented where the conjecture holds.