Extended fault-tolerant bipanconnectivity and panconnectivity of folded hypercubes

The folded hypercube ( n FQ for short) is a well-known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses. n FQ for any odd n is known to be bipartite. Let f be a faulty vertex in FQ_n , for n ≥ 2. In addition, let u and v be any two fault-free vertices in {f}. n FQ - f It has been shown that: 1) {f} n FQ - f contains a fault-free path P[u,v] of every length l with dFQ_n 2 2n 3 FQ_n d u v + ≤ l ≤ - and 2 | dFQ_n, FQ_n l - d u v where n ≥ 2; 2) {f} n FQ - f contains a fault-free P[u,v] path of every length l with n -1≤ l ≤ 2n - 3, where n ≥ 2 is even. In this paper, we extend the above-mentioned result to obtain two further properties as follows: 1) {f} n FQ - f contains a fault-free path P[u,v] of every length l with dFQ_n 2n 3 FQ_n d u v ≤ l ≤ - and 2 | dFQ_n FQ_n l - d u v for n ≥ 2 ; 2) {f} n FQ - f contains a fault-free path P[u,v] of every length l with n -1≤ l ≤ 2n - 2 for n ≥ 2 is even.