Random Induced Subgraphs of Generalizedn-Cubes

Abstract

The subject of this paper is to analyse connectivity and paths in random induced subgraphs of generalizedn-cubes over an alphabet A with |A|=α,Qαn. The vertices of Qαnaren-tuples (x1,…,xn), wherexi∈A,i=1,…,n, and two vertices are adjacent if they differ in exactly one coordinate. Random induced subgraphs, Γn<Qαn, are obtained by selecting each vertex of Qαnindependently with probability λn. The first theorem shows that for λn=(cln(n))/nthere exists a unique largest component in Γn<Qαnwhich contains almost all vertices and that the size of the second largest component is ≤Cn/ln(n),C>0. The second theorem describes connectivity and paths of Γnfor constant probability λ>0. It is proved that two verticesP,Qcontained in the largest component of Γn, that have distancekin Qα, have a distance ≤k+kin Γnand are typically connected by a large number of independent Γn-paths. Further, for any two verticesP,Qcontained in the largest Γncomponent there exists a constantcP,Q>0 such that their distance is ≤cP,Qn.