Shin-Shin Kao

Mutually independent Hamiltonian cycles in dual-cubes

AbstractThe hypercube family Qn is one of the most well-known interconnection networks in parallel computers. With Qn, dual-cube networks, denoted by DCn, was introduced and shown to be a (n+1)-regular, vertex symmetric graph with some fault-tolerant Hamiltonian properties. In addition, DCn’s are shown to be superior to Qn’s in many aspects. In this article, we will prove that the n-dimensional dual-cube DCn contains n+1 mutually independent Hamiltonian cycles for n≥2. More specifically, let vi∈V(DCn) for 0≤i≤|V(DCn)|−1 and let

On the Hamiltonian-Connectedness for Graphs Satisfying Ore's Theorem

\langle v_{0},v_{1},\ldots ,v_{|V(\mathit{DC}_{n})|-1},v_{0}\rangle

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

be a Hamiltonian cycle of DCn. We prove that DCn contains n+1 Hamiltonian cycles of the form

Deformed Honeycomb Tori

\langle v_{0},v_{1}^{k},\ldots,v_{|V(\mathit{DC}_{n})|-1}^{k},v_{0}\rangle

Super-connectivity and super-edge-connectivity for some interconnection networks

for 0≤k≤n, in which vik≠vik′ whenever k≠k′. The result is optimal since each vertex of DCn has only n+1 neighbors.

Spider web networks: a family of optimal, fault tolerant, hamiltonian bipartite graphs

Consider any undirected and simple graph G = (V,E), where V and E denote the vertex set and the edge set of G, respectively. Let |G| = |V| = n ≥ 3. The well-known Ore’s theorem states that if deg G (u) + deg G (v) ≥ n holds for each pair of nonadjacent vertices u and v of G, then G is hamiltonian. A similar theorem given by Erdos is as follows: if deg G (u) + deg G (v) ≥ n + 1 holds for each pair of nonadjacent vertices u and v of G, then G is hamiltonian-connected. In this paper, we improve both theorems by showing that any graph G satisfying the condition in Ore’s theorem is hamiltonian-connected unless G belongs to two exceptional families.

Mutually independent Hamiltonian cycles in alternating group graphs

Consider any undirected and simple graph G=(V, E), where V and E denote the vertex set and the edge set of G, respectively. Let |G|=|V|=n. The well-known Ores theorem states that if degG(u)+degG(v)≥n+k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k=−1, hamiltonian for k=0, and hamiltonian-connected for k=1. Lin et al. generalized Ores theorem and showed that under the same condition as above, G is r*-connected for 1≤r≤k+2 with k≥1. In this paper, we improve both theorems by showing that the hamiltonicity or r*-connectivity of any graph G satisfying the condition degG(u)+degG(v)≥n+k with k≥−1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem

Assume that m, n and s are integers with m⩾2, n⩾4, 0<s<n and s is of the same parity of m. The generalized honeycomb tori GHT (m,n,s) have been recognized as an attractive architecture to existing torus interconnection networks in parallel and distributed applications. Among the various families of graphs of GHT (m,n,s) numerous studies are devoted to honeycomb hexagonal torus HT (n) due to its nice symmetrical structure. Although each vertex of HT (n) is described by a three‐dimensional coordinate (x,y,z), the graph grows uniformly in the three directions. In this article, we propose a new class of graphs extended from HT (n), namely, deformed honeycomb torus DHT (h,l,r). DHT (h,l,r) is defined to allow the graph to grow in the three independent dimensions. We prove that this more general class of graphs still remains a subset of the generalized honeycomb torus. Furthermore, we have a concrete correspondence between any DHT (h,l,r) and the associated GHT (m,n,s).