The preclusion numbers and edge preclusion numbers in a class of Cayley graphs

Abstract

Let G be a hierarchical network (graph) with vertex set V(G) and edge set E(G). The preclusion set of the subnetwork G′ (defined as a smaller network but with the same topological properties as the original one) in G is a subset V′ of V(G) such that G−V′ has no any subnetwork isomorphic to G′. The preclusion number of G′ in G is F(G′)=min{|V′|:V′ is the preclusion set of G′}. Similarly, the edge preclusion set of G′ in G is a subset E′ of E(G) such that G−E′ has no any subnetwork isomorphic to G′. The edge preclusion number of G′ in G is f(G′)=min{|E′|:E′ is the edge preclusion set of G′}. The preclusion number and edge preclusion number are parameters which measure the fault tolerance of interconnection networks in the presence of failures. In this paper, we investigate the two parameters for a class of Cayley graphs T2n+1 generated by the transposition tree similar to the star. Let m be an integer with 0≤m≤n−1, let F(n,m)=F(T2(n−m)+1), and let f(n,m)=f(T2(n−m)+1). We prove that F(n,0)=f(n,0)=1, F(n,1)=2n(2n+1) with 2n+1 prime, f(n,1)≤4n(2n+1) with 2n+1 prime, F(n,n−1)=(2n+1)!6, f(n,n−1)=n(2n+1)!6, and 2n+12m(2m)!≤F(n,m)≤f(n,m)≤nm2n+12m(2m)!. Finally, we prove that F(n,m) is of order at most n3m−1.