How hard is deciding the existence of Red-Blue perfect matching?
As Mikhail noted, the problem was called Perfect Matching Cut in literatures. It was proved to be NP-complete in the following paper (see Theorem 1 on page 5), with a reduction from monotone 1-in-3-SAT:

Pinar Heggernes, Jan Arne Telle. Partitioning Graphs Into Generalized Dominating Sets.