# 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*(log^{2} *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*(log*n*) edges.

### Keywords

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