Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-17T21:32:13.891Z Has data issue: false hasContentIssue false

On the Cover Time for Random Walks on Random Graphs

Published online by Cambridge University Press:  01 September 1998

JOHAN JONASSON
Affiliation:
Chalmers University of Technology, S-412 96 Göteborg, Sweden, (e-mail: expect@math.chalmers.se)

Abstract

The cover time, C, for a simple random walk on a realization, GN, of [Gscr ](N, p), the random graph on N vertices where each two vertices have an edge between them with probability p independently, is studied. The parameter p is allowed to decrease with N and p is written on the form f(N)/N, where it is assumed that f(N)[ges ]c log N for some c>1 to asymptotically ensure connectedness of the graph. It is shown that if f(N) is of higher order than log N, then, with probability 1−o(1), (1−ε)N log N[les ]E[C[mid ]GN] [les ](1+ε)N log N for any fixed ε>0, whereas if f(N)=O(log N), there exists a constant a>0 such that, with probability 1−o(1), E[C[mid ]GN] [ges ](1+a)N log N. It is furthermore shown that if f(N) is of higher order than (log N)3 then Var(C[mid ]GN)= o((N log N)2) with probability 1−o(1), so that with probability 1−o(1), the stronger statement that (1−ε)N log N[les ]C[les ](1+ε)N log N holds.

Type
Research Article
Copyright
1998 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)