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Graphs of Non-Crossing Perfect Matchings

Abstract.

 Let P n be a set of n=2m points that are the vertices of a convex polygon, and let ℳ m be the graph having as vertices all the perfect matchings in the point set P n whose edges are straight line segments and do not cross, and edges joining two perfect matchings M 1 and M 2 if M 2=M 1−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of P n . We prove the following results about ℳ m : its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4.

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Received: October 10, 2000 Final version received: January 17, 2002

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ID="*" Partially supported by Proyecto DGES-MEC-PB98-0933

Acknowledgments. We are grateful to the referees for comments that helped to improve the presentation of the paper.

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Hernando, C., Hurtado, F. & Noy, M. Graphs of Non-Crossing Perfect Matchings. Graphs Comb 18, 517–532 (2002). https://doi.org/10.1007/s003730200038

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  • Key words. Perfect matching, Non-crossing configuration, Gray code