A finite simple graph is called a
k
-multicirculant if its automorphism group contains a cyclic semiregular subgroup having
k
orbits on the vertex set. It was shown by Giudici et al. that, if
k
is ...squarefree and coprime to 6, then a cubic arc-transitive
k
-multicirculant has at most
6
k
2
vertices (J. Combin. Theory Ser. B, 2017). In this paper, we classify the latter graphs under the assumption that their semiregular cyclic subgroups are contained in a soluble group of automorphisms acting transitively on the arc set of the graphs. As an application, cubic arc-transitive
p
-multicirculants are completely described for each odd prime
p
.
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