Abstract

We consider the cover time Eu [G], the expected time it takes a random walk that starts at u to visit all n vertices of a connected graph G. By combining a minimum weight spanning tree argument with a coupon collector argument, we obtain: E u [G] 1 2 (min T [W [T ]] + max v2G [D[u; v]]) where W [T ] is the weight of spanning tree T , and D[u; v] = H[u; v] \Gamma H[v; u] is the difference between the expected time it takes a random walk to hit v, and the expected time it takes it to afterwards get back to u. We use this bound to show: 1. maxG min u Eu [G] = (1 + o(1))2n 3 =27. This answers an open question of Aldous. 2. The n-path is the n-vertex tree on which the cover time is maximized. This confirms a conjecture of Brightwell and Winkler. 3. For regular graphs, Eu [G] ! 2n 2 . This improves the leading constant in previously known upper bounds. We also provide upper bounds on E + u [G], the expected time to cover G and return to u. 1 Introduction Let G be a connected u...

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