Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes

Choudum and Sunitha [Networks 40 (2002) 71–84] proposed the class of augmented cubes as a variation of hypercubes and showed that augmented cubes possess several embedding properties that the hypercubes and other variations do not possess. Recently, Hsu et al. [Information Processing Letters 101 (2007) 227–232] showed that the ndimensional augmented cube AQn, n 2, is weakly geodesic-pancyclic, i.e., for each pair of vertices u, v ∈ AQn and for each integer satisfying max{2d(u, v), 3} 2n where d(u, v) denotes the distance between u and v in AQn, there is a cycle of length that contains a u-v geodesic. In this paper, we obtain a stronger result by proving that AQn, n 2, is indeed geodesic-pancyclic, i.e., for each pair of vertices u, v ∈ AQn and for each integer satisfying max{2d(u, v), 3} 2n, every u-v geodesic lies on a cycle of length . To achieve the result, we first show that AQn − f , n 3, remains panconnected (and thus is also edge-pancyclic) if f ∈ AQn is any faulty vertex. The result of fault-tolerant panconnectivity is the best possible in the sense that the number of faulty vertices in AQn, n 3, cannot be increased.