# Construction of the digraph

(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 C1 and C2, 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 C1 and C2. 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 C1, C2, C3 and C4. The graph H(2) still has no directed 3-cycles.

(3) Let H(2)1, H(2)2, H(2)3and H(2)4 be 4 copies of H(2)of step (2). Let us denote the center cycle of Hi by Bi.

Now identify the flippable cycles Cj of 4 graphs H(2)i, for each j. The resulting graph H(3) has 4 flippable cycles Cj ( ) and 4 non-flippable cycles Bk ( ), where the cycles Cj, Bk 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 Ci(H(3)k) ( ) and non-flippable cycle Bi(H(3)k) which are mutually disjoint.

Now identify the cycle Ci(H(3)k) with Bi(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.