K4-expansions have the edge-Erdős-Pósa property

Long Cycles have the Edge-Erdős-Pósa Property

“…Then T together with a and the edges ab 1 , ab 2 , ab 3 forms a theta graph θ with branch vertex a. By Lemma 5, θ contains an even A-cycle that then is disjoint from z, a contradiction to (1).…”

“…Then C ∪Q is a theta graph such that two of its subdivided edges meet one of a i , a i+1 . Lemma 5 yields again an even A-cycle that avoids z, which is impossible by (1). For (ii) and (iii), suppose there were i < j and a C-path Q i with its endvertices in P i , and a C-path Q j with its endvertices in P j such that Q i and Q j meet, in a vertex v say.…”

“…Consider a cycle C ⊆ B through at least one vertex of A. Then, every vertex in A∩V (B) lies in C; otherwise we could find a C-path P in B that passes through some vertex a ′ ∈ V (B − C), which would imply with Lemma 5 that the theta graph C ∪P contained an even A-cycle that avoids z, which is impossible by (1 Let a 0 , a 1 , . .…”

“…Lemma 6 (Bruhn and Heinlein [1]). Let F be a class of graphs that has the Erdős-Pósa property with bounding function g. Let h : N → R be a function such that for every k and for every graph G that has a vertex z such that G − z does not contain any subgraph of F it holds that: either G contains k edgedisjoint subgraphs from F or there is an edge set F of size |F | ≤ h(k) that meets every subgraph from F .…”

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

“…There are not many results on the edge-Erdős-Pósa property, besides some further small results on H-models [2,4,9,12], it is only known that A-B-paths, A-paths and S-paths have the edge-Erdős-Pósa property but any of these paths such that its length is congruent to some x modulo some m do not have it [3].…”