Extended fault-tolerant cycle embedding in faulty hypercubes

We consider fault-tolerant embedding, where an-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 Q_n in 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 2ⁿ-2|F_v| in Q_n (n≥3) with |F_e| ≤ 2n-5, and |F_v|+|F_e|≤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.