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

Abstract

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=⌊n/2⌋ 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.