On the largest planar graphs with everywhere positive combinatorial curvature (extended arxiv version)

TL;DR

This paper proves that planar graphs with positive combinatorial curvature have a maximum of 208 vertices and faces with at most 41 sides, resolving a longstanding question in graph theory.

Contribution

It establishes upper bounds on the size and face complexity of planar PCC graphs, providing a complete answer to a question posed by DeVos and Mohar.

Findings

Maximum of 208 vertices in planar PCC graphs

Faces have at most 41 sides, and this bound is sharp

Refined discharging technique used in proof

Abstract

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.