Long $A$-$B$-paths have the edge-Erd\H os-Pósa property

Abstract

For a fixed integer ℓ a path is long if its length is at least ℓ. We prove that for all integers k and ℓ there is a number f (k, ℓ) such that for every graph G and vertex sets A, B the graph G either contains k edge-disjoint long A-B-paths or it contains an edge set F of size |F | ≤ f (k, ℓ) that meets every long A-B-path. This is the edge analogue of a theorem of Montejano and Neumann-Lara (1984). We also prove a similar result for long A-paths and long S-paths.

Erdős-Pósa from ball packing

“…: the cycles of length at least some given constant ℓ) and for K 4 -expansions [3]. There has also been much activity related to the edge-EP property for labeled graphs [16,2] and directed graphs [15]. From this account, one might gain the impression that the vertex-EP property and edge-EP property are essentially the same, and indeed sometimes the edge-EP property can be derived directly from its vertex variant [19].…”