Hamiltonian paths and Hamiltonian cycles passing through prescribed linear forests in star graph with fault-tolerant edges

• Published in: Discrete Applied Mathematics Vol. 334; pp. 68 - 80
• Main Authors: Xue, Shudan, Deng, Qingying, Li, Pingshan, Chen, Jianguo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 31.07.2023
• Subjects: Fault-tolerant edges; Hamiltonian cycle; Hamiltonian path; Prescribed Hamiltonian laceable; Prescribed linear forests; Star graph; Fault-tolerant edges; Hamiltonian cycle; Prescribed Hamiltonian laceable; Star graph; Prescribed linear forests; Hamiltonian path
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2023.02.016

In this paper, we focus on Hamiltonian paths and Hamiltonian cycles passing through prescribed linear forests in n-dimensional star graph Sn with fault-tolerant edges. Let F⊆E(Sn) be a fault-tolerant edges set and L be a linear forest of Sn−F, and |E(L)|+|F|≤n−3. For any two vertices u and v in different partite sets, we prove that if {u,v} and L are compatible in Sn, then there is a Hamiltonian path of Sn−F between u and v, which passes through each edge of L. That is, Sn is (n−3)-edge fault-tolerant prescribed Hamiltonian laceable for n≥4. And we show that there is a Hamiltonian cycle of Sn−F passing through each edge of L, that is, Sn is (n−3)-edge fault-tolerant prescribed Hamiltonian for n≥4. These results are optimal in the described sense.