The cover time of a biased random walk on a random regular graph of odd degree
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We consider a random walk process which prefers to visit previously unvisited edges, on the random r-regular graph G_r for any odd r≥ 3. We show that this random walk process has asymptotic vertex and edge cover times 1/r-2n n and r/2(r-2)n n, respectively, generalizing the result from Cooper, Frieze and Johansson from r = 3 to any larger odd r. This completes the study of the vertex cover time for fixed r≥ 3, with Berenbrink, Cooper and Friedetzky having previously shown that G_r has vertex cover time asymptotic to rn/2 when r≥ 4 is even.
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