Hamiltonian Properties of Faulty Recursive Circulant Graphs

Abstract

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

A parallel routing algorithm on recursive cube of rings networks employing Hamiltonian circuit Latin square

“…Later, we design a shortest path between two arbitrary nodes and then design a set of node-disjoint and shortest paths using the shortest path. Some papers previously published discussed conditions for availability of Hamiltonian circuit on a few graphs [11,16]. Instead of investigating conditions for the existence of a set of node-disjoint paths, we propose the parallel algorithm that constructs a set of k þ 1 node-disjoint paths on RCR(k, r, j) employing Hamiltonian circuit Latin square (HCLS).…”

Maximum induced subgraph of a recursive circulant

“…While retaining some attractive properties of hypercube (node-symmetry, highly recursive structure, maximum connectivity, simple shortest-path routing scheme, etc. ), RC(2 n , 4) achieves noticeable improvements in diameter [11], mean internode distance [11], pancyclicity and edgepancyclicity [2][3][4], and fault-tolerant hamiltonicity [13]. As a result, RC(2 n , 4) has been one potential interconnection network for multicomputer systems.…”

The super-connected property of recursive circulant graphs

“…1. Several interesting properties of G(2 m , 4) have been studied in the literature [3,[6][7][8]. For example, it was proved by Park and Chwa [7] that G(2 m , 4) is an m connected and Hamiltonian graph.…”

“…The diameter of G(2 m , 4) is less than that of Q m . G(2 m , 4) has good fault-tolerant Hamiltonian properties [8]. The super-connected property of G(2 m , 4) is important in such a sense that it can be considered as a measure of the reliability of G(2 m , 4).…”