Statement: If there exist two oriented cycles C and D on the cycle graph for a given permutation and none of the cycles leads to a valid 2-move, than it is always possible to find a (4,3)-sequence.

Proof: We prove the statement by listing all the possible configurations for two oriented cycles not allowing valid 2-moves. After that, we show a possible (4,3)-sequence. Below you can find the links for the (4,3)-sequences for each configuration. We also indicate what transposition was used in each case.

1 - (10 3 1 4 2)(9 7 5 8 6)
2 - (10 3 1 5 2)(9 7 4 8 6)
3 - (10 3 1 6 2)(9 7 4 8 5)
4 - (10 3 1 7 2)(9 6 4 8 5)
5 - (10 3 1 8 2)(9 6 4 7 5)
6 - (10 3 1 9 2)(8 6 4 7 5)
7 - (10 4 1 5 2)(9 7 3 8 6)
8 - (10 4 1 5 3)(9 7 2 8 6)
9 - (10 4 1 6 2)(9 7 3 8 5)
10 - (10 4 1 6 3)(9 7 2 8 5)
11 - (10 4 1 7 2)(9 6 3 8 5)
12 - (10 4 1 7 3)(9 6 2 8 5)
13 - (10 4 1 8 2)(9 6 3 7 5)
14 - (10 4 1 8 3)(9 6 2 7 5)
15 - (10 4 1 9 2)(8 6 3 7 5)
16 - (10 4 1 9 3)(8 6 2 7 5)
17 - (10 4 2 5 3)(9 7 1 8 6)
18 - (10 4 2 6 3)(9 7 1 8 5)
19 - (10 4 2 7 3)(9 6 1 8 5)
20 - (10 4 2 8 3)(9 6 1 7 5)
21 - (10 4 2 9 3)(8 6 1 7 5)
22 - (10 5 1 6 2)(9 7 3 8 4)
23 - (10 5 1 6 3)(9 7 2 8 4)
24 - (10 5 1 6 4)(9 7 2 8 3)
25 - (10 5 1 7 2)(9 6 3 8 4)
26 - (10 5 1 7 3)(9 6 2 8 4)
27 - (10 5 1 7 4)(9 6 2 8 3)
28 - (10 5 1 8 2)(9 6 3 7 4)
29 - (10 5 1 8 3)(9 6 2 7 4)
30 - (10 5 1 8 4)(9 6 2 7 3)
31 - (10 5 1 9 2)(8 6 3 7 4)
32 - (10 5 1 9 3)(8 6 2 7 4)
33 - (10 5 1 9 4)(8 6 2 7 3)
34 - (10 5 2 6 3)(9 7 1 8 4)
35 - (10 5 2 6 4)(9 7 1 8 3)
36 - (10 5 2 7 3)(9 6 1 8 4)
37 - (10 5 2 7 4)(9 6 1 8 3)
38 - (10 5 2 8 3)(9 6 1 7 4)
39 - (10 5 2 8 4)(9 6 1 7 3)
40 - (10 5 2 9 3)(8 6 1 7 4)
41 - (10 5 2 9 4)(8 6 1 7 3)
42 - (10 5 3 6 4)(9 7 1 8 2)
43 - (10 5 3 7 4)(9 6 1 8 2)
44 - (10 5 3 8 4)(9 6 1 7 2)
45 - (10 5 3 9 4)(8 6 1 7 2)
46 - (10 6 1 7 2)(9 5 3 8 4)
47 - (10 6 1 7 3)(9 5 2 8 4)
48 - (10 6 1 7 4)(9 5 2 8 3)
49 - (10 6 1 7 5)(9 4 2 8 3)
50 - (10 6 1 8 2)(9 5 3 7 4)
51 - (10 6 1 8 3)(9 5 2 7 4)
52 - (10 6 1 8 4)(9 5 2 7 3)
53 - (10 6 1 8 5)(9 4 2 7 3)
54 - (10 6 1 9 2)(8 5 3 7 4)
55 - (10 6 1 9 3)(8 5 2 7 4)
56 - (10 6 1 9 4)(8 5 2 7 3)
57 - (10 6 1 9 5)(8 4 2 7 3)
58 - (10 6 2 7 3)(9 5 1 8 4)
59 - (10 6 2 7 4)(9 5 1 8 3)
60 - (10 6 2 7 5)(9 4 1 8 3)
61 - (10 6 2 8 3)(9 5 1 7 4)
62 - (10 6 2 8 4)(9 5 1 7 3)
63 - (10 6 2 8 5)(9 4 1 7 3)
64 - (10 6 2 9 3)(8 5 1 7 4)
65 - (10 6 2 9 4)(8 5 1 7 3)
66 - (10 6 2 9 5)(8 4 1 7 3)
67 - (10 6 3 7 4)(9 5 1 8 2)
68 - (10 6 3 7 5)(9 4 1 8 2)
69 - (10 6 3 8 4)(9 5 1 7 2)
70 - (10 6 3 8 5)(9 4 1 7 2)
71 - (10 6 3 9 4)(8 5 1 7 2)
72 - (10 6 3 9 5)(8 4 1 7 2)
73 - (10 6 4 7 5)(9 3 1 8 2)
74 - (10 6 4 8 5)(9 3 1 7 2)
75 - (10 6 4 9 5)(8 3 1 7 2)
76 - (10 7 1 8 2)(9 5 3 6 4)
77 - (10 7 1 8 3)(9 5 2 6 4)
78 - (10 7 1 8 4)(9 5 2 6 3)
79 - (10 7 1 8 5)(9 4 2 6 3)
80 - (10 7 1 8 6)(9 4 2 5 3)
81 - (10 7 1 9 2)(8 5 3 6 4)
82 - (10 7 1 9 3)(8 5 2 6 4)
83 - (10 7 1 9 4)(8 5 2 6 3)
84 - (10 7 1 9 5)(8 4 2 6 3)
85 - (10 7 1 9 6)(8 4 2 5 3)
86 - (10 7 2 8 3)(9 5 1 6 4)
87 - (10 7 2 8 4)(9 5 1 6 3)
88 - (10 7 2 8 5)(9 4 1 6 3)
89 - (10 7 2 8 6)(9 4 1 5 3)
90 - (10 7 2 9 3)(8 5 1 6 4)
91 - (10 7 2 9 4)(8 5 1 6 3)
92 - (10 7 2 9 5)(8 4 1 6 3)
93 - (10 7 2 9 6)(8 4 1 5 3)
94 - (10 7 3 8 4)(9 5 1 6 2)
95 - (10 7 3 8 5)(9 4 1 6 2)
96 - (10 7 3 8 6)(9 4 1 5 2)
97 - (10 7 3 9 4)(8 5 1 6 2)
98 - (10 7 3 9 5)(8 4 1 6 2)
99 - (10 7 3 9 6)(8 4 1 5 2)
100 - (10 7 4 8 5)(9 3 1 6 2)
101 - (10 7 4 8 6)(9 3 1 5 2)
102 - (10 7 4 9 5)(8 3 1 6 2)
103 - (10 7 4 9 6)(8 3 1 5 2)
104 - (10 7 5 8 6)(9 3 1 4 2)
105 - (10 7 5 9 6)(8 3 1 4 2)
106 - (10 8 1 9 2)(7 5 3 6 4)
107 - (10 8 1 9 3)(7 5 2 6 4)
108 - (10 8 1 9 4)(7 5 2 6 3)
109 - (10 8 1 9 5)(7 4 2 6 3)
110 - (10 8 1 9 6)(7 4 2 5 3)
111 - (10 8 1 9 7)(6 4 2 5 3)
112 - (10 8 2 9 3)(7 5 1 6 4)
113 - (10 8 2 9 4)(7 5 1 6 3)
114 - (10 8 2 9 5)(7 4 1 6 3)
115 - (10 8 2 9 6)(7 4 1 5 3)
116 - (10 8 2 9 7)(6 4 1 5 3)
117 - (10 8 3 9 4)(7 5 1 6 2)
118 - (10 8 3 9 5)(7 4 1 6 2)
119 - (10 8 3 9 6)(7 4 1 5 2)
120 - (10 8 3 9 7)(6 4 1 5 2)
121 - (10 8 4 9 5)(7 3 1 6 2)
122 - (10 8 4 9 6)(7 3 1 5 2)
123 - (10 8 4 9 7)(6 3 1 5 2)
124 - (10 8 5 9 6)(7 3 1 4 2)
125 - (10 8 5 9 7)(6 3 1 4 2)
126 - (10 8 6 9 7)(5 3 1 4 2)