Among the hypothetical forms of fully covalent carbon lattices with either graphite-like sp(2) or mixed sp(2)/sp(3) bonding we investigated the possibility of monocrystals with a hollow structure. We have calculated the structural, elastic and electronic properties of two prototypical face-centered cubic lattices C-20 and C-22, respectively, making also a critical comparison to diamond and graphite. The first of these lattices belongs to a new class of sp(2)-bonded periodic solids which we call hollow graphites. We give a topological classification of such solids along with the algorithm to generate them. Both crystals, having a nanoporous lattice made of periodic sequence of adjacent cavities, rather than the tubular structure of schwarzites, are characterized by a large internal specific area and should be suitable to form reversible high capacity lithium insertion compounds.