  Two quantum spin models with bilinear-biquadratic exchange interactions are
constructed on the checkerboard lattice. It is proved that, under certain
sufficient conditions on the exchange parameters, their ground states consist
of two degenerate Shastry-Sutherland singlet configurations. The constructions
are studied for arbitrary spin-S. The sufficient conditions for the existence
of ferromagnetic ground state are also found exactly. The approximate quantum
phase diagrams are presented using the exact results, together with a
variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-1/2
ladder model, based on one of the above constructions, is considered which
admits exact solution for a large number of eigenstates. The ladder model is
shown to have exact level-crossing between the rung-singlet state and the AKLT
state in the singlet ground state. Also introduced is the notion of
perpendicularity for quantum spin vectors, which appears in the discussion on
one of the two checkerboard models, and is discussed in the Appendix.
