Popular matchings in complete graphs

Our input is a complete graph G on n vertices where each vertex has a strictranking of all other vertices in G. The goal is to construct a matching in G that is "globallystable" or popular. A matching M is popular if M does not lose a head-to-head election againstany matching M': here each vertex casts a vote forthe matching in {M,M'} in which it gets abetter assignment. Popular matchings need not exist in the given instance G and the popularmatching problem is to decide whether one exists or not. The popular matching problem in Gis easy to solve for odd n. Surprisingly, the problem becomes NP-hard for even n, as we showhere. This seems to be the first graph theoretic problem that is efficiently solvable when n hasone parity and NP-hard when n has the other parity.