In this note we show that there is a cubic graph of girth $5$ that is not a subgraph of any minimal Cayley graph. On the other hand, we show that any Generalized Petersen Graph $G(n,k)$ with $\gcd(n,k)=1$ is an induced subgraph of a minimal Cayley graph. These results give insights into two comments of László Babai in [L. Babai, \emph{Automorphism groups, isomorphism, reconstruction}. Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2, 1994].