A graph G is pancyclic if, for every 4 ÃÂÿ l ÃÂÿ |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 ÃÂÿ l ÃÂÿ |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 ÃÂÿ l ÃÂÿ |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 ÃÂÿ 3. They also asked if TQn is edge-pancyclic with (n-3) faults for n ÃÂÿ 3. We find that TQn is not edge-pancyclic with only one faulty edge for any n ÃÂÿ 3. Then we prove that TQn is node-pancyclic with (langle n/2rangle - 1) faulty edges for every n ÃÂÿ 3. The result is optimal in the sense that with langle n/2rangle faulty edges, the faulty TQn is not node-pancyclic for any n ÃÂÿ 3.
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机译：如果对于每4个γlγγ| V（G）|，G的长度为l的周期，则图G是全循环的。如果对于G的任意边e和每4Âl lƒÃ¢Â| | V（G）|，G的长度为l的周期，则图G是边泛循环的包含e。如果对于一个G的任意节点u和每4个Âllâ¢| V（G）|，G的长度为l的周期，则图G是节点全循环的包含你。扭曲的立方体是超立方体的重要变体。最近，范等人。证明n维扭曲立方体TQ n 对于每n个n边都是3边平移的。他们还问TQ n 是否是边平的？对于nƒ，具有（n-3）个断层的泛循环3。我们发现TQ n 不是边缘泛循环的，对于任何nƒ来说只有一个故障边缘，3。然后证明TQ n 是每n个n的（langle n / 2rangle-1）个故障边的节点全循环。3。结果是最优的从意义上讲，对于n个n / 2rangle的故障边缘，有故障的TQ n 对于任何nÂ3都不是节点全循环的。
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