Can someone please help to understand about the concept of "lifting Hamiltonian cycles (in Cayley graphs)"?
As an example how the existence of a Hamiltonian cycle is shown by using the concept of lifting Hamiltonian cycles in the Cayley graph of $\mathbb{Z}_7 \rtimes \mathbb{Z}_3$ (A Cayley graph with respect to a generating set $S$ of the form $S=\{u,t\}$, where $|u|=7, |t|=3$). ?
I have inserted an image of a Cayley graph of the semidirect product $\mathbb{Z}_7 \rtimes \mathbb{Z}_3$ under the above mentioned type of generating set. I have drawn it so that the 7-cycles and 3-cycles are clearly visible (Vertices lablelled 1-7, 8-14, 15-21 represent 7-cycles).
Is it possible to mention, with the aid of the labelling of the vertices a corresponding Hamiltonian cycle in the quotient graph that will be lifted and how the lifting occurs?
Please help with this question. Thanks a lot in advance.
In the paper of the following link they have employed this concept. In 3rd page it is mentioned as Marusic's method and they are thinking about a Hamiltonian cycle in the quotient Cayley graph and extending it to a cycle in the whole graph.
Then is it possible to help me to see and understand it from the example figure I have given.