Are the following Cayley digraphs Hamiltonian?

Question

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?

Answer 1

If I understood and coded correctly and sage didn't lie $A'G_4$ is not hamiltonian while $A'G_5$ is hamiltonian.

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Aded According to sage $A'G_6$ is hamiltonian.

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sage code:

def dicaygen(G,ge):
    """
    cayley digraph on gens $ge$
    """
    ve=[i for i in G]
    ed=[]
    for i in xrange(len(ve)):
        for j in xrange(len(ve)):
            if i==j:  continue

            a,b=ve[i],ve[j]
            if a*b^(-1)in ge:
                ed += [(a,b)]
    G=DiGraph(ed)
    return G

sage: n=4;an=AlternatingGroup(n);ge=[an([(1,k,2)]) for k in [ 3 .. n]];g=dicaygen(an,ge);g.is_hamiltonian()
False
sage: n=5;an=AlternatingGroup(n);ge=[an([(1,k,2)]) for k in [ 3 .. n]];g=dicaygen(an,ge);g.is_hamiltonian()
True

Answer 2

The answer to this question turns out to be already known, and can be found in the following paper: Cayley digraphs and (1,j,n)-sequencings of the alternating groups An.

As it turns out (Assuming I'm not misunderstanding the paper), $A'G_n$ is Hamiltonian for all $n \geq 3$, except when $n = 4$. (Which I didn't expect to be the answer.)

Thanks for your comments, they were very helpful!