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.
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments which improved the final version of the paper. Dong Ye: The author was partially supported by a grant from the Simons Foundation (No. 359516). Xiaoya Zha: Research supported by NSA Grant H98230-1-02192.
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This paper is dedicated to the memory of Victor Klee.
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Plummer, M.D., Ye, D. & Zha, X. Connectivity and \(W_v\)-Paths in Polyhedral Maps on Surfaces. Discrete Comput Geom 58, 217–231 (2017). https://doi.org/10.1007/s00454-017-9868-9
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DOI: https://doi.org/10.1007/s00454-017-9868-9