Symmetric property and edge-disjoint Hamiltonian cycles of the spined cube

The spined cube SQn, as a variant network of the hypercube Qn, was proposed in 2011 and has attracted much attention because of its smaller diameter. It is well-known that Qn is a Cayley graph. In the present paper, we show that SQn is an m-Cayley graph, that is its automorphism group has a semiregular subgroup acting on the vertices with m orbits, where m = 4 when n >= 6 and m = Ln/ 2 j when n <= 5 . Consequently, it shows that an SQn with n >= 6 can be partitioned into eight disjoint hypercubes of dimension n - 3 . As an application, it is proved that there exist two edge-disjoint Hamiltonian cycles in SQn when n >= 4 . Moreover, we prove that SQn is not vertex-transitive unless n <= 3. (c) 2023 Elsevier Inc. All rights reserved.