Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes

As an enhancement on the hypercube Qn, the augmented cube AQn, proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks, 40(2) (2002), 71–84], not only retains some of the favorable properties of Qn but also possesses some embedding properties that Qn does not. For example, AQn contains cycles of all lengths from 3 to 2n, but Qn contains only even cycles. In this paper, we obtain two stronger results by proving that AQn contains paths, between any two distinct vertices, of all lengths from their distance to 2n − 1; and AQn still contains cycles of all lengths from 3 to 2n when any (2n − 3) edges are removed from AQn. The latter is optimal since AQn is (2n − 1)-regular.