A Proof of the Molecular Conjecture

Authors:
Naoki Katoh;Shin-ichi Tanigawa
Affiliations:
Kyoto University, Department of Architecture and Architectural Engineering, Kyoto Daigaku Katsura, Nishikyo-ku, 615-8540, Kyoto, Japan;Kyoto University, Research Institute for Mathematical Sciences, Kitashirakawa-Oiwaketyo, Sakyo-ku, 606-8502, Kyoto, Japan
Venue:
Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
Year:
2011

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Abstract

We prove the Molecular Conjecture posed by Tay and Whiteley. This implies that a graph G can be realized as an infinitesimally rigid panel-hinge framework in ℝd by mapping each vertex to a rigid panel and each edge to a hinge if and only if $\bigl({d+1 \choose 2}-1\bigr)G$ contains ${d+1\choose2}$ edge-disjoint spanning trees, where $\bigl({d+1 \choose2}-1\bigr)G$ is the graph obtained from G by replacing each edge by $\bigl({d+1\choose2}-1\bigr)$ parallel edges.