On the rigidity of molecular graphs

Abstract

The rigidity of squares of graphs in three-space has important applications to the study of flexibility in molecules. The Molecular Conjecture, posed in 1984 by T.-S. Tay and W. Whiteley, states that the square G 2 of a graph G of minimum degree at least two is rigid if and only if G has six spanning trees which cover each edge of G at most five times. We give a lower bound on the degrees of freedom of G 2 in terms of forest covers of G. This provides a self-contained proof that the existence of the above six spanning trees is a necessary condition for the rigidity of G 2. In addition, we prove that the truth of the Molecular Conjecture would imply that our lower bound is tight, and would also imply that a conjecture of Jacobs on ‘independent’ squares is valid.

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Correspondence to Tibor Jordán.

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This work was supported by an International Joint Project grant from the Royal Society.

Supported by the MTA-ELTE Egerváry Research Group on Combinatorial Optimization and the Hungarian Scientific Research Fund grant no. T049671, T60802.

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Jackson, B., Jordán, T. On the rigidity of molecular graphs. Combinatorica 28, 645–658 (2008). https://doi.org/10.1007/s00493-008-2287-z

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Mathematics Subject Classification (2000)

  • 52C25
  • 05C90
  • 52B40