Conditional edge-fault pancyclicity of augmented cubes

• Conditional faulty means that each node is incident with at least two fault-free edges.
• Consider the augmented cube AQnAQn with fefe faulty edges under the conditional faulty.
• Prove that AQnAQn contains cycles of all lengths from 3 to 2n2n when fe⩽4n−12fe⩽4n−12.
• Improve the previous best result by Ma et al. in which fe⩽2n−3fe⩽2n−3.

The augmented cube AQnAQn, proposed by Choudum and Sunitha [7], is a variation of the hypercube QnQn and possesses many superior properties that the hypercube does not contain. In this paper, we show that, any n  -dimensional augmented cube with at most 4n−124n−12 faulty edges contains cycles of lengths from 3 to 2n2n under the condition that every node is incident with at least two fault-free edges, where n⩾3n⩾3. Ma et al. [21] obtained the same result but with the number of faulty edges up to 2n−32n−3. Our result improves Ma et al.ʼs result in terms of the number of fault-tolerant edges.