Node-Pancyclicity of Faulty Twisted Cubes

Abstract

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 TQn is edge-pancyclic for every n ≥ 3. They also asked if TQn is edge-pancyclic with (n−3) faults for n ≥ 3. We find that TQn is not edge-pancyclic with only one faulty edge for any n ≥ 3. Then we prove that TQn is node-pancyclic with (\langle n/2\rangle − 1) faulty edges for every n ≥ 3. The result is optimal in the sense that with \langle n/2\rangle faulty edges, the faulty TQn is not node-pancyclic for any n ≥ 3.