0 The cover time , the blanket time , and the Matthews bound

We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well known Matthews bounds on the cover time and prove that M/2 ≤ C = O(M (ln ln n) 2). We give a deterministic polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((ln ln n) 2), improving previous bound of O(ln n) of Matthews. The blanket time B was introduced by Winkler and Zuckerman: it is the expectation of the first time when all vertices are visited within a constant factor of number of times suggested by the stationary distribution. Obviously C ≤ B, and they conjectured B = O(C) and proved B = O(C ln n). Our bounds above are also valid for the blanket time, and so it follows that B = O(C(ln ln n) 2).