On the size of planar graphs with positive Lin-Lu-Yau Ricci curvature

TL;DR

This paper proves that planar graphs with minimum degree at least 3 and positive Lin-Lu-Yau Ricci curvature on every edge must have maximum degree at most 17, implying such graphs are finite, extending previous combinatorial curvature results.

Contribution

It establishes a bound on the maximum degree of planar graphs with positive Lin-Lu-Yau Ricci curvature, linking curvature conditions to graph finiteness and size.

Findings

Maximum degree of such graphs is at most 17

Graphs with positive Ricci curvature are finite

Extension of combinatorial curvature results to Ricci curvature

Abstract

We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $\Delta(G)\leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar [{\em Trans. Amer. Math. Soc., 2007}] on the size of planar graphs with positive combinatorial curvature.