Monotone Paths in Planar Convex Subdivisions
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(logΔ 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.
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