We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We ask: for which hereditary classes of graphs is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs, a class that includes examples of interest in the statistical setting.
In REF , it is proved that the largest hereditary class of bipartite graphs for which the reconfiguration graph of perfect matchings with flips is connected is the class of chordal bipartite graphs.
On the Switch Markov Chain for Perfect Matchings
JACM
Mathematics Computer Science