The flip Markov chain for connected regular graphs

Mahlmann and Schindelhauer (2005) defined a Markov chain which they called kFlipper, 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 ε of uniform in total variation distance after poly(n, r, log(ε−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-stablising protocol in which nodes spontaneously perform random flips in order to repair the network. An earlier version of this paper appeared as an extended abstract in PODC 2009 [6]. This research was partly performed while the first three authors were visiting the Simons Institute for the Theory of Computing. Supported by EPSRC Research Grant EP/M004953/1. Supported by the Australian Research Council Discovery Project DP140101519. Supported by EPSRC Research Grant EP/D00232X/1.