Planar Graphs have Independence Ratio at least 3/13

TL;DR

This paper proves that every planar graph has an independence ratio of at least 3/13, improving previous bounds and providing a new lower limit independent of the 4 Color Theorem.

Contribution

The authors establish a new lower bound of 3/13 for the independence ratio in planar graphs, surpassing the previous bound of 2/9, without relying on the 4 Color Theorem.

Findings

Improved the lower bound on independence ratio to 3/13

Demonstrated the bound is independent of the 4 Color Theorem

Strengthened understanding of independence sets in planar graphs

Abstract

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$. In 1968, Erd\H{o}s asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.