Non-revisiting paths on surfaces with low genus
A revisiting path to a face of a graph of a polytope, is a path that leaves the face at a vertex \(x\), and revisits at a vertex \(y\) (with no intersection with the face in between). The non-revisiting path conjecture says that any two vertices of a polytope \(P\) can be joined by a path that does not revisit any face of \(P\). The authors have previously proved this false for all orientable surfaces of genus \(g\geq 2\) and all nonorientable surfaces of nonorientable genus \(h\geq 4\). Here it is shown by an elementary proof that the conjecture is true for the sphere, projective plane, torus and Klein bottle, only leaving the case of the connected sum of three copies of the projective plane still open.
scientific article; zbMATH DE number 1121502
scientific article