(0)
Let *C* be a directed *n*-cycle
,
and *v* and *w* two vertices
outside of *C* with edges
and
those
.
We call such a graph
a double-wheel. Note that the edges
and
are non-flippable
without making directed 3-cycles, while the edges
are flippable.

(1)
Let *B* be a directed cycle of size 4.
Let us add 2 double-wheels whose directed cycles
*C*_{1} and *C*_{2}, respectively, are size 4 as
Figure 1.
Then the resulting graph *H*^{(1)} has all the edges
non-flippable except the eight edges in the two directed
cycles *C*_{1} and *C*_{2}.
Note that the graph *H*^{(1)} has no directed 3-cycles.

(2)
As same as step (1), we add a pair of double-wheels to
each flippable cycle as Figure 2.
The resulting graph *H*^{(2)} has all the edges
non-flippable except 16 edges in 4 directed cycles.
Let us call the center cycle *B* and 4 flippable
cycles *C*_{1}, *C*_{2}, *C*_{3} and *C*_{4}.
The graph *H*^{(2)} still has no directed 3-cycles.

(3)
Let
*H*^{(2)}_{1},
*H*^{(2)}_{2},
*H*^{(2)}_{3}and
*H*^{(2)}_{4} be 4 copies of *H*^{(2)}of step (2). Let us denote the center cycle of *H*_{i} by *B*_{i}.

Now identify the flippable cycles *C*_{j} of 4 graphs
*H*^{(2)}_{i},
for each *j*.
The resulting graph *H*^{(3)} has 4 flippable
cycles *C*_{j} (
)
and 4 non-flippable
cycles *B*_{k} (
), where the cycles
*C*_{j}, *B*_{k} are mutually disjoint.
Other edges are all non-flippable.
Observe that this identification does not make no directed
3-cycles.

(4)
Let
*H*^{(3)}_{1},
*H*^{(3)}_{2},
*H*^{(3)}_{3}
and
*H*^{(3)}_{4} be *n* ()
copies of *H*^{(3)} of
step (3), and each graph
*H*^{(3)}_{k} have 4 flippable
cycles
*C*_{i}(*H*^{(3)}_{k}) (
)
and
non-flippable cycle
*B*_{i}(*H*^{(3)}_{k}) which are
mutually disjoint.

Now identify the cycle
*C*_{i}(*H*^{(3)}_{k}) with
*B*_{i}(*H*^{(3)}_{k+1}) for each fixed pair (*i*, *j*, *k*),
where the summation is taken in modulo *n*.
The resulting graph *G* has no directed 3-cycles and
no flippable edges.