On the switch Markov chain for perfect matchings

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

Applications of matching theory in constraint programming

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Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures