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](http://mathoverflow.net/questions/112661/spanning-trees-of-plane-graphs) 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?