The cover time of a biased random walk on a random cubic graph

Our aim in this paper is to analyse a variation on a simple random walk that may tend to speed up the cover time of a connected graph. This variation is just one of several possible approaches which include (i) non-bactracking walks, see Alon, Benjamini, Lubetzky and Sodin [3], (ii) walks that prefer unused edges, see Berenbrink, Cooper and Friedetzky [4] or (iii) walks that a biassed toward low degree vertices, see Cooper, Frieze and Petti [7] or any number of other ideas. In this paper we study idea (ii) in the context of random regular graphs of odd degree, solving a problem left from [4].