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 $n\geq 8$ and $r = r(n)\geq 2$. Formally, we prove that the distribution of the flip chain will be within $\varepsilon$ of uniform in total variation distance after $\text{poly}(n,r,\log(\varepsilon^{-1}))$ 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.