Sandwich Theorem of Cover Times

Yilun Shang SUTD-MIT International Design Center, Singapore University of Technology and Design, Singapore 138682 Correspondence should be addressed to Yilun Shang; shylmath@hotmail.com Received 26 June 2013; Accepted 22 July 2013 Academic Editors: I. Beg, P. E. Jorgensen, V. Makis, and O. Pons Copyright © 2013 Yilun Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on a connection between cover times of graphs and Talagrand’s theory of majorizing measures, we establish sandwich theorems for cover times as well as blanket times.


Introduction
Let = ( , ) be a connected graph with vertices and edges. Consider a simple random walk on : we start at a vertex V ∈ ; if at step we are at vertex , then we move from to a neighbor at step + 1 with probability −1 , where is the degree of . Let V be the expectation operator governing the random walk started at vertex V ∈ . The cover time of is defined as where ( ) is the first time ≥ 1 that all vertices of have been traversed [1]. Another relevant quantity is the strong -blanket time [2]. For V ∈ , let V = V /2 be the stationary measure of the random walk, and let V, be the number of visits to V up to time . For ∈ (0, 1), the strong -blanket time is defined as where ( ) is the first time ≥ 1 such that holds for any two vertices and V. Clearly, we have ( , ) ≥ ( ) for any ∈ (0, 1). We refer the reader to [1,3] for more background information on random walks. Recently, Ding et al. [4] established an important connection between cover times (blanket times) and the 2 functional from Talagrand's theory of majorizing measures [5,6]; see Theorem 1. We first review the 2 functional in brief. Consider a compact metric space ( , ), and let 0 = 1, = 2 2 for ≥ 1. For a partition P of and an element ∈ , denote by P( ) the unique set ∈ P containing . An admissible sequence {A } ≥0 of partitions of is that A +1 is a refinement of A , and |A | ≤ for ≥ 0. The 2 functional is defined as where diam( ) represents the diameter of , that is, diam( ) = sup , ∈ ( , ). Throughout the paper, we view graph as a metric space with distance induced by the commute time ( , V) between two vertices , V ∈ . Hence, ( , ) is a compact metric space.
A comparison theorem for cover times is also presented.
Theorem 2 (see [4]). Suppose that graphs and are on the same vertex set , and and are the distances induced by respective commute times. If there exists a number ≥ 1 such that ( , V) ≤ ⋅ ( , V) for all , V ∈ , then In this paper, we extend this nice comparison theorem and provide several applications.

Results
We have the following results.
Theorem 3. Suppose that three graphs , 1 , and 2 are on the same vertex set and that , 1 , and 2 are the distances induced by respective commute times. If there exist ∈ (0, 1], 1 > 0, and 2 > 0 such that 1 Proof. From the assumption, we have for all vertices and V. Capitalizing on the -inequality, which says that where = 1 if ∈ (0, 1] and = −1 if ∈ (1, ∞), we obtain by using definition (4).
It follows from Theorem 1 that as desired.
Generally, the conditions imposed on commute times in Theorem 3 are thorny, if possible, to test, especially for complex and large-scale graphs. However, commute times for recursive graphs are likely to be estimated (see, e.g., [7]).
Based on this comparison characterization, we have the following bounds regarding the ratio of cover times.
where ( , V) is the expected hitting time from to V in 1 .
Proof. If 1 = 2 , we obtain An upper bound of cover time [8] yields where = | | as mentioned before.
The following result is a "sandwich theorem" for monotonic graph sequences.

Corollary 5.
Maintaining the notations of Theorem 3, if 1 is a graph obtained by deleting an edge from and 2 is obtained by adding an edge to , one has where = | | is the number of edges in .
Proof. Since 1 and 2 have − 1 and + 1 edges, respectively, we obtain by [3, Theorem 2.9] that for any , V ∈ . Thus, the result follows directly from Theorem 3 by taking = 1.
We mention that it is recently shown in [9] that ( 2 )/4 ≤ ( ). We conclude the paper with a result on -strong blanket times analogous to our main theorem. Proof. The same proof in Theorem 3 applies by using Theorem 1.