On some super fault-tolerant Hamiltonian graphs

Y-Chuang Chen, Chang-Hsiung Tsai, Lih-Hsing Hsu, Jimmy J.M. Tan
2004 Applied Mathematics and Computation  
A k-regular Hamiltonian and Hamiltonian connected graph G is super fault-tolerant Hamiltonian if G remains Hamiltonian after removing at most k À 2 nodes and/or edges and remains Hamiltonian connected after removing at most k À 3 nodes and/or edges. A super fault-tolerant Hamiltonian graph has a certain optimal flavor with respect to the fault-tolerant Hamiltonicity and Hamiltonian connectivity. In this paper, we investigate a construction scheme to construct super fault-tolerant Hamiltonian
more » ... phs. In particularly, twisted-cubes, crossed-cubes, and M€ o obius cubes are all special cases of this construction scheme. Therefore, they are all super fault-tolerant Hamiltonian graphs. A path is a sequence of adjacent edges ðv 0 ; v 1 Þ; ðv 1 ; v 2 Þ; . . . ; ðv mÀ1 ; v m Þ, written as For our purpose in this paper, a path may contain only one node. A path is a Hamiltonian path if its nodes are distinct and they span V . A cycle is a path with at least three nodes such that the first node is the same as the last one. A cycle is a Hamiltonian cycle if it traverses every node of G exactly once. A graph G is Hamiltonian if it has a Hamiltonian cycle, and G is Hamiltonian connected if there exists a Hamiltonian path joining any two nodes of G. The architecture of an interconnection network is usually represented by a graph. There are a lot of mutually conflicting requirements in designing the topology of interconnection networks. It is almost impossible to design a network which is optimum for all conditions. One has to design a suitable network depending on the requirements of their properties. The Hamiltonian property is one of the major requirements in designing the topology of networks. Fault tolerance is also desirable in massive parallel systems that have relatively high probability of failure. There are many researches on the ring embedding problems in faulty interconnection networks [2,10,
doi:10.1016/s0096-3003(02)00933-5 fatcat:yyjcdvvrbbctfohdntgrron4ay