Conditional (edge-)fault-tolerant strong Menger (edge) connectivity of folded hypercubes

Qiao and Yang (2017) proved that all n-dimensional folded hypercubes are (2n−2)-conditional edge-fault-tolerant strongly Menger edge connected for n≥5. Yang, Zhao and Zhang (2017) showed that all n-dimensional folded hypercubes are (2n−3)-conditional fault-tolerant strongly Menger connected for n≥8. In this paper, we improve the result of Qiao and Yang by showing that all n-dimensional folded hypercubes are (3n−5)-conditional edge-fault-tolerant strongly Menger edge connected for n≥5. Moreover, we present an example to show that our result is optimal with respect to the maximum tolerated edge faults. In addition, we show that the result of Yang, Zhao and Zhang is optimal by proving that the n-dimensional folded hypercubes are not (2n−2)-conditional fault-tolerant strongly Menger connected for n≥8.