$K_4$-subdivisions have the edge-Erdős-Pósa property

Long Cycles have the Edge-Erdős-Pósa Property

“…Thus, from now on, we may assume that 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).…”

“…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 .…”

“…an edge set) of size |X| ≤ f (k) such that G − X is devoid of subgraphs from F . Then not only have cycles both the ordinary and the edge-Erdős-Pósa property but this is also true for even cycles [6,14,4]; for A-cycles 1 [10,11], cycles that each contain at least one vertex from some fixed set A; long cycles [13,2], cycles that have a certain pre-fixed minimum length; K 4 -subdivisions [13,1]; and many other classes of graphs.…”

Erdős-Pósa from ball packing

“…Since then, a new wave of papers has demonstrated the edge-EP property for θ r -expansions [8] (where θ r is the multigraph consisting of two vertices joined by r parallel edges), for long cycles [6] (i.e. : 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].…”