Distances in Random Induced Subgraphs of Generalized n-Cubes

Abstract

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.