Consider the Cayley graphs $A'G_n$ on the alternating group $A_n$ with generating set $S = \{(1i2) : 3 \leq i \leq n\}$, for $n \geq 4$.

See the following page on Alternating Group Graphs for information on the undirected versions, $AG_n$: Alternating Group Graph. It is known that the $AG_n$ is Hamiltonian for all $n\geq3$.

I suspect that $A'G_n$ is not Hamiltonian, for all $n \geq 4$. It certainly isn't for $n = 4$, which I simply checked by hand.

These graphs, were they to be non-Hamiltonian for all $n \geq 3$, would give a nice infinite family of counterexamples to show that the Lovász Conjecture doesn't hold for Cayley digraphs. According to the linked Wikipedia page, various counterexamples were found by R.A. Rankin, although no mention is made as to what those counterexamples are.

So the question is, is $A'G_n$ Hamiltonian?