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Bipartite Rigidity

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Combinatorial Methods in Topology and Algebra

Part of the book series: Springer INdAM Series ((SINDAMS,volume 12))

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Abstract

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general 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 a bipartite graph is planar only if its balanced shifting does not contain K 3, 3. We also discuss potential applications of this theory to Jockusch’s cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes.

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Notes

  1. 1.

    Author Eran Nevo reflecting a joint work in progress with Gil Kalai and Isabella Novik.

  2. 2.

    Relation (2) asserts that the velocities respect (infinitesimally) the distance along an embedded edge. If these relations apply to all pairs of vertices the velocities necessarily come from a rigid motion of the entire space.

  3. 3.

    In higher dimensions the class of shifted complexes is much reacher than the class of threshold complexes.

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Acknowledgements

Research was partially supported by Marie Curie grant IRG-270923 and ISF grant 805/11.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel

    Eran Nevo

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  1. Eran Nevo
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Correspondence to Eran Nevo .

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Editors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, Florida, USA

    Bruno Benedetti

  2. Département de Mathématiques, Université de Fribourg, Fribourg, Switzerland

    Emanuele Delucchi

  3. Université Paris-Diderot, , IMJ-PRG, Paris 7, Paris, France

    Luca Moci

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Nevo, E. (2015). Bipartite Rigidity. In: Benedetti, B., Delucchi, E., Moci, L. (eds) Combinatorial Methods in Topology and Algebra. Springer INdAM Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-20155-9_20

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