  Using a graph approach to quantum systems, we prove that descriptions of
3-dim Kochen-Specker (KS) setups as well as descriptions of 3-dim spin systems
by means of Greechie lattices that we find in the literature are wrong. Correct
lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert
subspace equations are given. To enable exhaustive generations of 3-dim KS
setups by means of recently found "stripping technique," bipartite graph
generation is used to provide us with lattices with equal numbers of elements
and blocks (orthogonal triples of elements) - up to 41 of them. We obtain
several new results on such lattices and hypergraphs, in particular on
properties such as superposition and orthoraguesian equations. Since a
bipartite graph approach has recently been applied to CSS
(Calderbank-Shor-Steane) and graph states on the one hand, and span programs,
quantum walks, and quantum search on the other, our results also enable the
study of these quantum information fields by means of hypergraphs and lattices.
