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 with n vertices. Previous upper bounds for general graphs of O(1VI jE[)~ll and O(I Vl I-EWdmin) 12) imply an upper bound of O(n2). We show an upper bound on general graphs of 0(3 IEI log IV]), which implies an upper bound of O(n log 2 n). The previous lower bound was f2 ( Ig l log lVI) for trees, t21 In our main result, we show a lower bound of Q(IVI (logam,~ [Vt) 2) for trees, which yields a lower bound of Q(nlog2n). We also extend our techniques to show an upper bound for general graphs of O(max{E= Ti} log I VI ).