This paper focuses on the embeddability of hypercubes in an important class of Cayley graphs, known as augmented cubes. An n-dimensional augmented cube AQₙ is constructed by augmenting the n-dimensional hypercube Qₙ with additional edges, thus making Qₙ a spanning subgraph of AQₙ. Dong and Wang (2019) first posed the problem of determining the number of Qₙ-isomorphic subgraphs in AQₙ, which still remains open. By exploiting the Cayley properties of AQₙ, we establish a lower bound for this number. What's more, we develop a method for constructing pairs of Qₙ-isomorphic subgraphs in AQₙ with the minimum number of common edges. This is accomplished through the use of reciprocal perfect matchings, a technique that also relies on the Cayley property of AQₙ. As an application, we prove that AQₙ admits n-1 edge-disjoint Hamiltonian cycles when n3 is odd and n-2 cycles when n is even, thereby confirming a conjecture by Hung (2015) for the odd case. Additionally, we prove that AQₙ has a fault-free cycle of every even length from 4 to 2ⁿ with up to 4n-8 faulty edges, when each vertex is incident to at least two fault-free edges. This result not only provides an alternative proof for the fault-tolerant Hamiltonicity of established by Hsieh and Cian (2010), but also extends their work by demonstrating the fault-tolerant bipancyclicity of AQₙ.
Yang et al. (Thu,) studied this question.