Extended Fault-Tolerant Cycle Embedding in Faulty Hypercubes

We consider fault-tolerant embedding, where an n-dimensional faulty hypercube, denoted by Q(n), acts as the host graph, and the longest fault-free cycle represents the guest graph. Let F(v) be a set of faulty nodes in Q(n). Also, let F(e) be a set of faulty edges in which at least one end-node of each edge is faulty, and let F(e) be a set of faulty edges in which the end-nodes of each edge are both fault-free. An edge in Q(n) is said to be critical if it is either fault-free or in F(e). In this paper, we prove that there exists a fault-free cycle of length at least 2(n) - 2 vertical bar F(v)vertical bar in Q(n) (n >= 3) with vertical bar F(e)vertical bar <= 2n - 5, and vertical bar F(v)vertical bar + vertical bar F(e)vertical bar <= 2n - 4, in which each node is incident to at least two critical edges. Our result improves on the previously best known results reported in the literature, where only faulty nodes or faulty edges are considered.