Frames, $A$-paths and the Erd\H{o}s-P\'osa property

TL;DR

This paper extends the frame technique from Erdős-Pósa theorem to A-paths, providing new proofs and verifying properties for long and even A-paths, while showing limitations for edge versions.

Contribution

It adapts the frame method to A-paths, offers a simplified proof of Gallai's theorem, and establishes Erdős-Pósa properties for specific A-path classes.

Findings

Verified Erdős-Pósa property for long A-paths

Verified Erdős-Pósa property for even A-paths

Established that even A-paths lack the edge-Erdős-Pósa property

Abstract

A key feature of Simonovits' proof of the classic Erd\H{o}s-P\'osa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique to $A$-paths. With it we deduce a simple proof of Gallai's theorem, although with a worse bound, and we verify the Erd\H{o}s-P\'osa property for long and for even $A$-paths. We also show that even $A$-paths do not have the edge-Erd\H{o}s-P\'osa property.