Just look up the definitions of your problem. So as an example, to find a Hamiltonian cycle, you want to find a tour which visits each vertex once and which starts and ends in the same vertex. So somehow you need to pass through all the vertices in the outer cycle, as well as the vertices in the inner cycle, without passing through a vertex twice. So basically you can walk over all the vertices in the outer cycle, than move to the inner cycle, walk over the inner cycle and return to the outer cycle. At least, this should be possible because you state that all vertices of the outer graph are connected with all vertices in the inner graph?
Ps. usually you talk about cycle's and not about circles afaik.