Collecting coupons on trees, and the cover time of random walks

Abstract

We consider the cover timeE _u [G], the expected time it takes a random walk that starts at vertexu to visit alln vertices of a connected graphG. Aleliunas et al introduced the spanning tree argument: for any spanning treeT of the graphG, E _u [G]≤W[T], whereW[T] is the sum of commute times along the edges ofT. By refining the spanning tree argument we obtain:

$$E_u [G] \leqslant \frac{1}{2}(\mathop {\min }\limits_T [W[T]] + \mathop {\max }\limits_{\upsilon \in G} [H[u,\upsilon ] - H[\upsilon ,u]])$$

whereH[u,v] is the hitting time fromu tov. We use this bound to show:

• 1. max _G min _u E _u [G]=(1+o(1))2n ³/27. This answers an open question of Aldous.

• 2. Then-path is then-vertex tree on which the cover time is maximized. This confirms a conjecture of Brightwell and Winkler.

• 3. For regular graphs,E _u [G]<2n ². This improves the leading constant in previously known upper bounds.

We also provide upper bounds onE ⁺ _u [G], the expected time to coverG and return tou.