Nonrevisiting Paths on Surfaces

AbstraeL The nonrevisiting path conjecture for polytopes, which is equivalent to the Hirsch conjecture, is open. However, for surfaces, the nonrevisiting path conjecture is known to be true for polyhedral maps on the sphere, projective plane, toms, and a Klein bottle. Barnette has provided counterexamples on the orientable surface of genus 8 and nonorientable surface of genus 16. In this note the question is settled for all the remaining surface except the connected sum of three copies of the projective plane.