The flip Markov chain for connected regular graphs

Mahlmann and Schindelhauer (2005) defined a Markov chain which they called -Flipper, and showed that it is irreducible on the set of all connected regular graphs of a given degree (at least 3). We study the 1-Flipper chain, which we call the flip chain, and prove that the flip chain converges rapidly to the uniform distribution over connected -regular graphs with vertices, where and . Formally, we prove that the distribution of the flip chain will be within of uniform in total variation distance after steps. This polynomial upper bound on the mixing time is given explicitly, and improves markedly on a previous bound given by Feder et al. (2006). We achieve this improvement by using a direct two-stage canonical path construction, which we define in a general setting.

This work has applications to decentralised networks based on random regular connected graphs of even degree, as a self-stabilising protocol in which nodes spontaneously perform random flips in order to repair the network.