Christofides algorithm (by hand)
In fact, you have been correct all along. However, you have not finished Christofides algorithm yet.

The next step is "Calculate Euler tour".
In the last picture, you have obtained a multigraph all of whose vertices have even degrees. So it must have an Eulerian path. For example, let us select the Eulerian path A -> F -> A -> C -> B -> D -> E -> D -> A.

The last step is "Remove repeated vertices, giving the algorithm's output".
Removing the second A and the second D on that above path, we obtain a wanted output, A -> F -> C -> B -> D -> E -> A. Done.



Please note it is not very meaningful to apply Christofides algorithm on the given graph. 

"The Christofides algorithm is an algorithm for finding approximate solutions to the travelling salesman problem, on instances where the distances form a metric space (they are symmetric and obey the triangle inequality". The given distances do not obey the triangle inequality, since d(B,D) + d(D, E) = 1 + 4 &lt; 6 = d(B,E). Moreover, the (direct) distance between A and B is not given, raising doubt whether this is a travelling salesman problem at all.