An
m
-Cayley graph
Γ
over a group
G
is defined as a graph which admits
G
as a semi-regular group of automorphisms with
m
orbits. This generalises the notions of a Cayley graph (where
m
=
1
) and a
...bi-Cayley graph
(where
m
=
2
). The
m
-Cayley graph
Γ
over
G
is said to be
normal
if
G
is normal in the automorphism group
Aut
(
Γ
)
of
Γ
, and
core-free
if the largest normal subgroup of
Aut
(
Γ
)
contained in
G
is the identity subgroup. In this paper, we investigate properties of symmetric
m
-Cayley graphs in the special case of valency 3, and use these properties to develop a computational method for classifying connected cubic core-free symmetric
m
-Cayley graphs. We also prove that there is no 3-arc-transitive normal Cayley graph or bi-Cayley graph (with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31–41
2005
). Using our classification method, we give a new proof of the fact that there are exactly 15 connected cubic core-free symmetric Cayley graphs, two of which are Cayley graphs over non-abelian simple groups. We also show that there are exactly 109 connected cubic core-free symmetric bi-Cayley graphs, 48 of which are bi-Cayley graphs over non-abelian simple groups, and that there are 1, 6, 81, 462 and 3267 connected cubic core-free 1-arc-regular 3-, 4-, 5-, 6- and 7-Cayley graphs, respectively.
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