Spanning trees of plane graphs containing an edge of every face

A triangulation has a spanning tree with the required property if and only if its dual graph has a hamiltonian path (is traceable).

Zamfirescu constructed a 3-regular 3-connected planar non-traceable graph on 88 vertices. The dual of this graph is a triangulation with no spanning tree with required properties.

a reference: Tudor Zamfirescu, Three small cubic graphs with interesting Hamiltonian properties, Journal of Graph Theory, Vol. 4 (1980), 287-292.