Hamiltonian Cycle Problem

What you are looking for is a Hamilton cycle decomposition of the complete graph $K_n$, for odd $n$.

An example of how this can be done (among many other results in the area) is given in: D. Bryant, Cycle decompositions of complete graphs, in Surveys in Combinatorics, vol. 346, Cambridge University Press, 2007, pp. 67–97.

For odd $n$, let $n=2r+1$, take $\mathbb{Z}_{2r} \cup \{\infty\}$ as the vertex set of $K_n$ and let $D$ be the orbit of the $n$−cycle

\[(\infty, 0, 1, 2r − 1, 2, 2r − 2, 3, 2r − 3,\ldots , r − 1, r + 1, r)\]

under the permutation $\rho_{2r}$ [Here $\rho_{2r}=(0,1,\ldots,2r-1)$]. Then $D$ is a decomposition of $K_n$ into $n$-cycles.

Here is the starter cycle for a Hamilton cycle decomposition of $K_{13}$, given in the paper:

If you rotate the starter, you obtain the other Hamilton cycles in the decomposition.

The method of using a "starter" cycle under the action of a cyclic automorphism is typical in graph decomposition problems.