Meander graphs

Abstract

We consider aMarkov chain Monte Carlo approach to the uniform sampling of meanders. Combinatorially, a meanderM = [A : B] is formed by two noncrossing perfect matchings, above A and below B the same endpoints, which form a single closed loop. We prove that meanders are connected under appropriate pairs of balanced local moves, one operating on A and the other on B. We also prove that the subset of meanders with a fixed B is connected under a suitable local move operating on an appropriately defined meandric triple in A. We provide diameter bounds under such moves, tight up to a (worst case) factor of two. The mixing times of the Markov chains remain open. © 2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.