Sarbazi-Azad, Ould-Khaoua, and Mackenzie proved in 2001 that there exists a Hamiltonian cycle in a pyramid network and they also constructed a Hamiltonian path between apex and each of 4 frontiers of a pyramid network. The fault tolerance is a crucial matter for parallel computing, especially in a large network. This work improves Sarbazi-Azad et al.’s result and considers other relative problems in pyramid networks such as the fault tolerant Hamiltonian problem and the Hamiltonian-connected problem. The problem of finding Hamiltonian cycles in a pyramid network with one faulty node (link) is investigated. Additionally, the Hamiltonian-connectedness of a pyramid network can be shown by constructing a Hamiltonian path between any two distinct nodes in it.