Monotone Paths in Planar Convex Subdivisions

Abstract

Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most Δ edges. Then, starting from every vertex there is a path with at least Ω(log_Δ n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivision of the plane into n convex faces where exactly k faces are unbounded. Then, there is a path with at least Ω(log(n/k)/loglog(n/k)) edges that is monotone in some direction. This bound is also the best possible.

In 3-space, we show that for every n ≥ 4, there exists a polytope P with n vertices, bounded vertex degrees, and triangular faces such that every monotone path on the 1-skeleton of P has at most O(log² n) edges. We also construct a polytope Q with n vertices, and triangular faces, (with unbounded degree however), such that every monotone path on the 1-skeleton of Q has at most O(logn) edges.

Preliminary results were reported by the authors in [3,6]. Dumitrescu was supported in part by the NSF grant DMS-1001667; Rote was supported in part by the Centre Interfacultaire Bernoulli in Lausanne and by the National Science Foundation; Tóth was supported in part by the NSERC grant RGPIN 35586. Research by Tóth was conducted at the Fields Institute in Toronto and at the Centre Interfacultaire Bernoulli in Lausanne.