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

TL;DR

This paper investigates the $W_v$-Path Conjecture in polyhedral maps on surfaces, establishing conditions under which vertices are connected by paths that do not revisit facets, and relating this to surface topology and local connectivity.

Contribution

It proves that if three internally disjoint $(x,y)$-paths are homotopic, then a $W_v$-path exists, and establishes bounds on local connectivity needed for $W_v$-paths based on surface topology.

Findings

Existence of $W_v$-paths under homotopy conditions

Bounds on local connectivity for $W_v$-paths depending on surface

Sharp bounds for the sphere case

Abstract

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.