Spanning Trees in Locally Planar Triangulations

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 23g + 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 23g + 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.