The flip Markov chain for connected regular graphs

Mahlmann and Schindelhauer (2005) defined a Markov chain which they called k-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 2r-regular graphs with n vertices, where n≥ 8 and r = r(n)≥ 2. Formally, we prove that the distribution of the flip chain will be within varepsilon of uniform in total variation distance after poly(n,r,log(varepsilon-1)) 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.