Nonrevisiting paths on surfaces
The non-revisiting path conjecture for a polyhedral map would claim that between any two of its vertices there exists an edge-path which, once it leaves a face, never revisits it. This conjecture is known to be true for such maps on the sphere, projective plane, torus and Klein bottle. \textit{D. W. Barnette} [Geom. Dedicata 46, No. 2, 211-214 (1993; Zbl 0777.52007)] gave counter-examples on the orientable surface of genus 8, and the non-orientable surface of genus 16. In this paper, the authors provide counter-examples for all surfaces except for the first four, and the connected sum of three copies of the projective plane, for which the problem remains open.
scientific article; zbMATH DE number 881220