On the largest planar graphs with everywhere positive combinatorial curvature (extended arxiv version)
Luca Ghidelli · arXiv (Cornell University) · 2017
A planar PCC graph is a simple connected planar graph with everywhere positive combinatorial curvature which is not a prism or an antiprism and with all vertices of degree at least 3. We prove that every planar PCC graph has at most 208 vertices, thus answering completely a question raised by DeVos and Mohar. The proof is based on a refined discharging technique and on an accurate low-scale combinatorical description of such graphs. We also prove that all faces in a planar PCC graph have at most 41 sides, and this result is sharp as well.