Mathematische Zeitschrift

, Volume 235, Issue 2, pp 315–334

Monotone paths on polytopes

  • Christos A. Athanasiadis
  • Paul H. Edelman
  • Victor Reiner
Original article

DOI: 10.1007/s002090000152

Cite this article as:
Athanasiadis, C., Edelman, P. & Reiner, V. Math Z (2000) 235: 315. doi:10.1007/s002090000152

Abstract.

We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytopeP with respect to a generic functionalf. The third author has conjectured that this graph is always (d\(-1\))-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-connected for any d-polytope with \(d \geq 3\). However, we disprove the conjecture in general by exhibiting counterexamples for each \(d \geq 4\) in which the graph has a vertex of degree two.

We also re-examine the Baues problem for cellular strings on polytopes, solved by Billera, Kapranov and Sturmfels. Our analysis shows that their positive result is a direct consequence of shellability of polytopes and is therefore less related to convexity than is at first apparent.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Christos A. Athanasiadis
    • 1
  • Paul H. Edelman
    • 2
  • Victor Reiner
    • 2
  1. 1.Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden (e-mail: athana@math.kth.se)SE
  2. 2.School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA (e-mail: edelman@math.umn.edu; reiner@math.umn.edu)US