Switch-based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

11/19/2020

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by   Pieter Kleer, et al.

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We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on n vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least n/2+7 can be transformed into each other by switch operations of size at most 10, implying that the switch Markov chain using switches of size at most 10 is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.