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

TL;DR

This paper analyzes the cover time of a biased random walk on odd-degree regular graphs, establishing asymptotic formulas for vertex and edge cover times that extend previous results to all odd degrees.

Contribution

It generalizes the asymptotic cover time results for biased random walks from degree 3 to all odd degrees r ≥ 3 on regular graphs.

Findings

Vertex cover time asymptotic to (1/(r-2)) n log n

Edge cover time asymptotic to (r/(2(r-2))) n log n

Completes the understanding of cover times for odd-degree regular graphs

Abstract

We consider a random walk process which prefers to visit previously unvisited edges, on the random $r$-regular graph $G_r$ for any odd $r\geq 3$. We show that 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 any larger 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 $r\geq 4$ is even.