Bipancyclicity in k-Ary n-Cubes with Faulty Edges under a Conditional Fault Assumption

We prove that a k-ary 2-cube Q_2^k with three faulty edges but where every vertex is incident with at least two healthy edges is bipancyclic, if k\ge 3, and k-pancyclic, if k\ge 5 is odd (these results are optimal). We go on to show that when k\ge 4 is even and n\ge 3, any k-ary n-cube Q_n^k with at most 4n-5 faulty edges so that every vertex is incident with at least two healthy edges is bipancyclic, and that this result is optimal.