It is well known that, given "n" red points and "n" blue points on acircle, it is not always possible to find a plane geometric. Hamiltonian alternating path. In this work we prove that if we relax the constraint on the path from being plane to being 1-plane, then the problem always has a solution, and even a Hamiltonian alternating cycle can be obtained on all instances. we also extend this kind of result to other configurations and provide remarks on similar problems.