On Cayley representations of finite graphs over abelian p-groups

We construct a polynomial-time algorithm which given a graph $\mathrm{\Gamma }$ finds the full set of non-equivalent Cayley representations of $\mathrm{\Gamma }$ over the group $D\cong C_p\times C_{p^k}$, where $p\in \{2,3\}$ and $k\ge 1$. This result implies that the recognition and the isomorphism problems for Cayley graphs over $D$ can be solved in polynomial time.