A sufficient condition for the existence of plane spanning trees on   geometric graphs
  Let P be a set of n > 2 points in general position in the plane and let G be a geometric graph with vertex set P. If the number of empty triangles uvw in P for which the subgraph of G induced by {u,v,w} is not connected is at most n-3, then G contains a non-self intersecting spanning tree. 
Eduardo Rivera-Campo and Virginia Urrutia-Galicia
math.CO cs.CG cs.DM