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:
whereH[u,v] is the hitting time fromu tov. We use this bound to show:
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max G min u E u [G]=(1+o(1))2n 3/27. This answers an open question of Aldous.
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Then-path is then-vertex tree on which the cover time is maximized. This confirms a conjecture of Brightwell and Winkler.
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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.
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Feige, U. Collecting coupons on trees, and the cover time of random walks. Comput Complexity 6, 341–356 (1996). https://doi.org/10.1007/BF01270386
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DOI: https://doi.org/10.1007/BF01270386