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.