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Bipartite Rigidity

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers $k,l$ the notions of $(k,l)$-rigid and $(k,l)$-stress free bipartite graphs. This theory coincides with the study of Babson--Novik's balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, and gluing lemmas, and apply these results to derive a bipartite analog of the rigidity criterion for planar graphs. Our result asserts that for a planar bipartite graph $G$ its balanced shifting, $G^b$, does not contain $K_{3,3}$; equivalently, planar bipartite graphs are generically $(2,2)$-stress free. We also discuss potential applications of this theory to Jockusch's cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes.