Consider a planar graph $G$ which is a triangulation.

Is it possible to find a two-colorable subgraph $H$ of $G$ which has a common edge with every face of $G$?

It is known that it is not always possible to take $H$ to be a spanning tree. See this MO question.

Note that we do not require $H$ to be connected.

If it is true, is there an efficient algorithm to find such a subgraph?