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We consider a random walk process which prefers to visit previously unvisitededges, on the random $r$-regular graph $G_r$ for any odd $r\geq 3$. We showthat this random walk process has asymptotic vertex and edge cover times$\frac{1}{r-2}n\log n$ and $\frac{r}{2(r-2)}n\log n$, respectively,generalizing the result from Cooper, Frieze and Johansson from $r = 3$ to anylarger odd $r$. This completes the study of the vertex cover time for fixed$r\geq 3$, with Berenbrink, Cooper and Friedetzky having previously shown that$G_r$ has vertex cover time asymptotic to $\frac{rn}{2}$ when $r\geq 4$ iseven.

The cover time of a biased random walk on a random regular graph of odd degree