, Volume 235, Issue 2, pp 315-334

Monotone paths on polytopes

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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.

Received April 6, 1999 / in final form October 1, 1999 / Published online July 20, 2000