Fault-Tolerant Hamiltonicity of Twisted Cubes

The Fault-Tolerant Hamiltonian Problems of Crossed Cubes with Path Faults

“…For example, it has more cycles than the hypercube [15], and it can embed binary trees [22], paths of odd and even lengths [14], [16], and many-to-many disjoint path covers [25]. The crossed cube has received many researchers' attention [4], [5], [12], [14], [19], [28]. Its definition will be introduced in the next section.…”

“…The faulttolerant Hamiltonicity of a Hamiltonian graph G indicates the maximum integer k such that G − F remains Hamiltonian for every F ⊂ (V(G) ∪ E(G)) with |F| ≤ k. Similarly, the fault-tolerant Hamiltonian connectedness of a Hamiltonian connected graph G indicates the maximum integer k such that G − F remains Hamiltonian connected for every F ⊂ (V(G) ∪ E(G)) with |F| ≤ k [19]. In other words, a graph G is k-fault-tolerant Hamiltonian (respectively, kfault-tolerant Hamiltonian connected) if it remains HamilCopyright c 2015 The Institute of Electronics, Information and Communication Engineers tonian (respectively, Hamiltonian connected) after removing at most k vertices and/or edges.…”

“…In other words, a graph G is k-fault-tolerant Hamiltonian (respectively, kfault-tolerant Hamiltonian connected) if it remains HamilCopyright c 2015 The Institute of Electronics, Information and Communication Engineers tonian (respectively, Hamiltonian connected) after removing at most k vertices and/or edges. Huang et al [19] have shown that the n-dimensional crossed cube is (n − 2)-faulttolerant Hamiltonian and (n − 3)-fault-tolerant Hamiltonian connected if n ≥ 3.…”

Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

“…Embedding paths and cycles in various well-known networks, such as the hypercube and some well-known variations of the hypercube, have been extensively investigated in the literature (see, e.g., Tsai [5] for the hypercubes, Fu [6] for the folded hypercubes, Huang et al [7] and Yang et al [8] for the crossed cubes, Yang et al [9] for the twisted cubes, Hsieh and Chang [10] for the Möbius cubes, Li et al [11] for the star graphs and Xu and Ma [12] for a survey on this topic). Recently, Cao et al [13] have shown that every edge of is contained in cycles of every length from 4 to 2 except 5, and every pair of vertices with distance is connected by paths of every length from to 2 − 1 except 2 and 4 if = 1, from which contains a Hamilton cycle for ⩾ 2 and a Hamilton path between any pair of vertices for ⩾ 3.…”

Embedding a Mesh into a Crossed Cube

“…[3,6,7,8,9,11,13]. 따 라서 최단경로, 헤밀톤 경로, 사이클, 트리, 메쉬 등을 교차큐 브에 임베딩하는 연구도 활발히 진행되었다 [3,6,12,19,14,10,5]. …”