If the test is:

4 6
1 2
1 3
1 4
2 3
2 4
3 4

then all created routes are as follows:

1 - 2 - 3 - 1
1 - 2 - 4 - 1
1 - 3 - 4 - 1
2 - 3 - 4 - 2
1 - 2 - 3 - 4 - 1
1 - 2 - 4 - 3 - 1
1 - 3 - 2 - 4 - 1

Am I right?

Indeed I was right, the answer for this test is 6.

Can this problem be solved faster than in O(2^n * n^3)?