  The cover time C of a graph G is the expected time for a random walk starting
from the worst vertex to cover all vertices in G. Similarly, the blanket time B
is the expected time to visit all vertices within a constant factor of number
of times suggested by the stationary distribution. (Our definition will be
slightly stronger than this.) Obviously, all vertices are covered when the
graph is blanketed, and hence C <= B. The blanket time is introduced by Winkler
and Zuckerman motivated by applications in Markov estimation and distributed
computing. They conjectured B =O(C) and proved B=O(C ln n ). In this paper, we
introduce another parameter M motivated by Matthews' theorem and prove
  M/2 <= C <= B = O(( M ln ln n)^2).
  In particular, B = O(C (ln ln n)^2). The lower bound is still valid for the
cover time C(\pi) starting from the stationary distribution. We also show that
there is a polynomial time algorithm to approximate M within a factor of 2 and
so does for C within a factor of O((ln ln n)^2), improving previous bound of
O(ln n) of Matthews'.
