Chun-Nan Hung

Fault-Tolerant Hamiltonicity of Twisted Cubes

The twisted cube TQn, is derived by changing some connection of hypercube Qn according to specific rules. Recently, many topological properties of this variation cube are studied. In this paper, we consider a faulty twisted n-cube with both edge and/or node faults. Let F be a subset of V(TQn)?E(TQn), we prove that TQn?F remains hamiltonian if |F|?n?2. Moreover, we prove that there exists a hamiltonian path in TQn?F joining any two vertices u, v in V(TQn)?F if |F|?n?3. The result is optimum in the sense that the fault-tolerant hamiltonicity (fault-tolerant hamiltonian connectivity respectively) of TQn is at most n?2 (n?3 respectively).

Ring embedding in faulty pancake graphs

In this paper, we consider the fault Hamiltonicity and the fault Hamiltonian connectivity of the pancake graph Pn. Assume that F ⊆ V (Pn) ∪ E (Pn). For n ≥ 4, we prove that Pn - F is Hamiltonian if |F|≤(n - 3) and Pn - F is Hamiltonian connected if |F|≤(n - 4). Moreover, all the bounds are optimal.

On the construction of combined k‐fault‐tolerant Hamiltonian graphs

A graph G G is a combined kk-fault-tolerant Hamiltonian graph (also called a combined kk-Hamiltonian graph) if G F is Hamiltonian for every subset F (V V(G) [ [ E(G G)) with |F F |= kk. A combined kk-Hamiltonian graph G with |V V(G)| = n is optimal if it has the minimum number of edges among all nn-node k-Hamiltonian graphs. Using the concept of node expansion, we present a powerful construction scheme to construct a larger combined k-Hamiltonian graph from a given smaller graph. Many previous graphs can be constructed by the concept of node expansion. We also show that our construction maintains the optimality property in most cases. The classes of optimal combined k-Hamiltonian graphs that we constructed are shown to have a very good diameter. In particular, those optimal combined 1Hamiltonian graphs that we constructed have a much smaller diameter than that of those constructed previously by Mukhopadhyaya and Sinha, Harary and Hayes,

Optimal 1-Hamiltonian graphs

In this paper, we present a family of 3-regular, planar, and hamiltonian graphs. Any graph in this family remains hamiltonian if any node or any edge is deleted. Moreover, the diameter of any graph in this family is O(√p) where p is the number of nodes.

Christmas tree: a versatile 1-fault-tolerant design for token rings

Abstract The token ring topology is required in token passing approach used in distributed operating systems. Fault tolerance is also required in the designs of distributed systems. Note that 1 -fault-tolerant design for token rings is equivalent to design of 1 -Hamiltonian graphs. This paper introduces a new family of graphs called Christmas tree, denoted by CT(s) . The graph CT(s) is a 3 -regular, planar, 1 -Hamiltonian, and Hamiltonian-connected graph. The number of nodes in CT(s) is 3·2 s −2 . Its diameter is 1 if s=1 , 3 if s=2 , and 2s if s≥3 .

Embedding Hamiltonian paths and Hamiltonian cycles in faulty pancake graphs

The use of pancake and star networks as an interconnection network has been studied by many researchers. The fault tolerance for Hamiltonian networks is also an important issue. In this paper, we prove that an n-dimensional faulty pancake graph contains a Hamiltonian cycle with |F| /spl les/ n - 3 faults. Furthermore, there exist Hamiltonian paths between two arbitrary but distinct nodes in a faulty pancake graph with |F| /spl les/ n - 4 faults.

On the isomorphism between cyclic-cubes and wrapped butterfly networks

We show that the cyclic-cubes defined by Ada W.C. Fu and S.C. Chau (1998) are isomorphic to k-ary wrapped butterfly networks.

The correct diameter of trivalent Cayley graphs

Abstract Let n be a positive integer with n≥2 . The trivalent Cayley interconnection network, denoted by TCIN(n) , is proposed by Vadapalli and Srimani (1995). Later, Vadapalli and Srimani (1996) claimed that the diameter of TCIN(n) is 2n−1 . In this paper, we argue that the above claim is not correct. Instead, we show that the diameter of TCIN(n) is 2n−1 only for n=2 and 2n−2 for all other cases.

The most vital edges of matching in a bipartite graph

Let G = (V,E) be an undirected graph having an edge weight we ≥ 0 for each e ϵ E. An edge is called a most vital edge (with respect to weighted matching) if its removal from G results in the largest decrease in the total weight of the maximum weighted matching. In this paper, we study the most vital edges of matching in a weighted bipartite graph. We present an O(n3) algorithm to obtain the most vital edges.

Strong Fault-Hamiltonicity for the Crossed Cube and Its Extensions

Fault-Hamiltonicity is an important measure of robustness for interconnection networks. Given a graph G = (V, E). The goal is to ensure that G − F remains Hamiltonian for every F ⊆ V ∪E such that |...