In this paper we study distances in random subgraphs of a generalized n-cube
[Qscr ]ns over
a finite alphabet S of size s.
[Qscr ]ns is the direct product of complete graphs over s vertices,
its vertices being the n-tuples (x1, …, xn), with xi ∈
S, i = 1, … n, and two vertices being
adjacent if they differ in exactly one coordinate. A random (induced) subgraph γ
of
[Qscr ]ns
is obtained by selecting
[Qscr ]ns-vertices with independent probability pn and then inducing the
corresponding edges from
[Qscr ]ns. Our main result is that dγ
(P,Q) [les ] [2k+3]d[Qscr ]ns
(P,Q) almost
surely for P,Q ∈ γ, pn = n−a and 0 [les ] a < ½, where k =
[1+3a/1−2a]
and dγ and d[Qscr ]ns
denote the distances in γ and [Qscr ]ns, respectively.