Connectivity and $$W_v$$Wv-Paths in Polyhedral Maps on Surfaces

The $$W_v$$Wv-path conjecture due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch conjecture. Klee proved that the $$W_v$$Wv-path conjecture is true for all 3-polytopes (3-connected plane graphs), and conjectured even more, namely that the $$W_v$$Wv-path conjecture is true for all general cell complexes. This general $$W_v$$Wv-path conjecture was verified for polyhedral maps on the projective plane and the torus by Barnette, and on the Klein bottle by Pulapaka and Vince. Let G be a graph polyhedrally embedded in a surface $$\Sigma $$Σ, and x, y be two vertices of G. In this paper, we show that if there are three internally disjoint (x, y)-paths which are homotopic to each other, then there exists a $$W_v$$Wv-path joining x and y. For every surface $$\Sigma $$Σ, define a function $$f(\Sigma )$$f(Σ) such that if for every graph polyhedrally embedded in $$\Sigma $$Σ and for a pair of vertices x and y in V(G), the local connectivity $$\kappa _G(x,y) \ge f(\Sigma )$$κG(x,y)≥f(Σ), then there exists a $$W_v$$Wv-path joining x and y. We show that $$f(\Sigma )=3$$f(Σ)=3 if $$\Sigma $$Σ is the sphere, and for all other surfaces $$3-\tau (\Sigma )\le f(\Sigma )\le 9-4\chi (\Sigma )$$3-τ(Σ)≤f(Σ)≤9-4χ(Σ), where $$\chi (\Sigma )$$χ(Σ) is the Euler characteristic of $$\Sigma $$Σ, and $$\tau (\Sigma )=\chi (\Sigma )$$τ(Σ)=χ(Σ) if $$\chi (\Sigma )< -1$$χ(Σ)<-1 and 0 otherwise. Further, if x and y are not cofacial, we prove that G has at least $$\kappa _G(x,y)+4\chi (\Sigma )-8$$κG(x,y)+4χ(Σ)-8 internally disjoint $$W_v$$Wv-paths joining x and y. This bound is sharp for the sphere. Our results indicate that the $$W_v$$Wv-path problem is related to both the local connectivity $$\kappa _G(x,y)$$κG(x,y), and the number of different homotopy classes of internally disjoint (x, y)-paths as well as the number of internally disjoint (x, y)-paths in each homotopy class.