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

The $W_v$-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 $W_v$-Path Conjecture is true for all 3-polytopes (3-connected plane graphs), and conjectured even more, namely that the $W_v$-Path Conjecture is true for all general cell complexes. This general $W_v$-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 $\Sigma$, 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 $W_v$-path joining $x$ and $y$. For every surface $\Sigma$, define a function $f(\Sigma)$ such that if for every graph polyhedrally embedded in $\Sigma$ and for a pair of vertices $x$ and $y$ in $V(G)$, the local connectivity $\kappa_G(x,y) \ge f(\Sigma)$, then there exists a $W_v$-path joining $x$ and $y$. We show that $f(\Sigma)=3$ if $\Sigma$ is the sphere, and for all other surfaces $3-\tau(\Sigma)\le f(\Sigma)\le 9-4\chi(\Sigma)$, where $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$, and $\tau(\Sigma)=\chi(\Sigma)$ if $\chi(\Sigma)< -1$ and 0 otherwise. Further, if $x$ and $y$ are not cofacial, we prove that $G$ has at least $\kappa_G(x,y)+4\chi(\Sigma)-8$ internally disjoint $W_v$-paths joining $x$ and $y$. This bound is sharp for the sphere. Our results indicate that the $W_v$-path problem is related to both the local connectivity $\kappa_G(x,y)$, 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.