Frames, $A$-paths and the Erdős-Pósa property

Abstract

A key feature of Simonovits' proof of the classic Erdős-Pósa 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 Erdős-Pósa property for long and for even $A$-paths. We also show that even $A$-paths do not have the edge-Erdős-Pósa property.

16 pages. A number of small issues fixed, a table with results on paths and cycles added