How do I do the change from an Eulerian tour to a Hamiltonian cycle in the Christofides algorithm? In the original paper from Christofides (1976)1 I found that
A Hamiltonian circuit $i$ of $G$ can be found with cost $$C(i_H) \le C(T^*) + C(M_0^*)<\frac32C(i^*)$$ where $C(i^*)$ is the optimal value of the TSP tour $i^*$.
On cs.stackexchange.com I found this answer:
However, we wanted a Hamilton cycle (another name for a TSP tour). The idea now is to follow the Eulerian tour. Whenever we're supposed to visit a vertex we have already encountered, we just "skip" this edge. Eventually the tour will reach a new vertex, and then we just connect the previous vertex with the new one.
Are there not more than one possible Hamiltonian cycle in one Eulerian tour? Is it good enough to take just some Hamiltonian cycle? Does this already satisfy the upper bound of $3/2$?
 Christofides, N. (1976). Worst-case analysis of a new heuristic for the travelling salesman problem. Carnegie-Mellon Univ Pittsburgh Pa Management Sciences Research Group.