Improved Erd\H{o}s-P\'osa inequalities for odd cycles in planar graphs

TL;DR

This paper proves that in planar graphs, the ratio between the minimum vertex set hitting all odd cycles and the maximum number of disjoint odd cycles is at most 4, improving previous bounds.

Contribution

The paper establishes a tighter upper bound of 4 for the Erdős-Pósa ratio for odd cycles specifically in planar graphs, advancing understanding of cycle packings and transversals.

Findings

Erdős-Pósa ratio in planar graphs is at most 4

Improves previous bound of 6

Provides new insights into odd cycle packings in planar graphs

Abstract

In an undirected graph, the odd cycle packing number is the maximum number of pairwise vertex-disjoint odd cycles. The odd cycle transversal number is the minimum number of vertices that hit every odd cycle. The maximum ratio between transversal and packing number is called Erd\H{o}s-P\'osa ratio. We show that in planar graphs, this ratio does not exceed 4. This improves on the previously best known bound of 6 by Kr\'al', Sereni and Stacho.