Graphs of Non-Crossing Perfect Matchings

Let Pn be a set of n m points that are the vertices of a convex polygon and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross and edges joining two perfect matchings M and M if M M a b c d a d b c for some points a b c d of Pn We prove the following results about Mm its diameter is m it is bipartite for every m the connectivity is equal to m it has no Hamilton path for m odd m and nally it has a Hamilton cycle for every m even m