In this paper, we consider the fault Hamiltonicity, and the fault Hamiltonian connectivity of the (n, k)-star graph S n,k . Assume that F ʲ V(S n,k ) ʴ E(S n,k ). For n ؊ k ≥ 2, we prove that S n,k ؊ F is Hamiltonian if ͦFͦ ≤ n ؊ 3 and S n,k ؊ F is Hamiltonian connected if ͦFͦ ≤ n ؊ 4. For n ؊ k ‫؍‬ 1, S n,n؊1 is isomorphic to the n-star graph S n which is known to be Hamiltonian if and only if n > 2 and Hamiltonian connected if and only if n ‫؍‬ 2. Moreover, all the bounds are tight.

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We study some topological and algorithmic properties of a recently defined hierarchical interconnection network, the hierarchical crossed cube HCC(k, n), which draws upon constructions used within the well-known hypercube and also the crossed cube. In particular, we study: the construction of shortest paths between arbitrary vertices in HCC(k, n); the connectivity of HCC(k, n); and one-to-all broadcasts in parallel machines whose underlying topology is HCC(k, n) (with both one-port and multi-port store-and-forward models of communication). Moreover, some of our proofs are applicable not just to hierarchical crossed cubes but to hierarchical interconnection networks

A k-containerC(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) is a k*-container if every vertex of G is incident with a path in C(u, v). A graph G is k*-connected if there exists a k*-container between any two distinct vertices u and v. A k-regular graph G is super spanning connected if G is i*-connected for all 1 ≤ i ≤ k. In this paper, we prove that the (n, k)-star graph Sn,k is super spanning connected if n ≥ 3 and (n-k) ≥ 2.