Spanning trees in locally planar triangulations (Q1333338)

Thomassen has recently shown that a triangulation of the orientable surface of genus \(g\) has a spanning tree of maximum degree at most 4, provided every noncontractible cycle has length at least \(2^{3g+4}\). We show that a 4-connected triangulation of the orientable surface of genus \(g\) has a spanning tree of maximum degree at most 3, provided that every noncontractible cycle has length at least \(2^{3g+5}\). This proves a result suggested by Thomassen. Examples demonstrate that some condition on the length of the noncontractible cycles is necessary for a result of this kind.