Node-Pancyclicity of Faulty Twisted Cubes

A graph G is pancyclic if, for every 4 ¿ l ¿ |V (G)|, G has a cycle of length l. A graph G is edge-pancyclic if, for an arbitrary edge e of G and every 4 ¿ l ¿ |V(G)|, G has a cycle of length l containing e. A graph G is node-pancyclic if, for an arbitrary node u of G and every 4 ¿ l ¿ |V (G)|, G has a cycle of length l containing u. The twisted cube is an important variant of the hypercube. Recently, Fan et al. proved that the n-dimensional twisted cube TQ_n is edge-pancyclic for every n ¿ 3. They also asked if TQ_n is edge-pancyclic with (n-3) faults for n ¿ 3. We find that TQ_n is not edge-pancyclic with only one faulty edge for any n ¿ 3. Then we prove that TQ_n is node-pancyclic with (langle n/2rangle - 1) faulty edges for every n ¿ 3. The result is optimal in the sense that with langle n/2rangle faulty edges, the faulty TQ_n is not node-pancyclic for any n ¿ 3.