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.