Covering times of random walks on bounded degree trees and other graphs

The motivating problem for this paper is to find the expected covering time of a random walk on a balanced binary tree withn vertices. Previous upper bounds for general graphs ofO(|V| |E|)(1) andO(|V| |E|/dmin)(2) imply an upper bound ofO(n2). We show an upper bound on general graphs ofO(Δ |E| log |V|), which implies an upper bound ofO(n log2n). The previous lower bound was Ω(|V| log |V|) for trees.(2) In our main result, we show a lower bound of Ω(|V| (log d max |V|)2) for trees, which yields a lower bound of Ω(n log2n). We also extend our techniques to show an upper bound for general graphs ofO(max{EπTi} log |V|).