The Edge-Erdős-Pósa Property

Bruhn, Henning; Heinlein, Matthias; Joos, Felix

doi:10.1007/s00493-020-4071-7

Cited by 4 publications

(12 citation statements)

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“…Using the fact that long A-paths have the edge-Erdős-Pósa property, we can also prove that long cycles have this property. This has already been proven in [5], but we give a shorter proof (although with a worse hitting set function). A 1-vertex-hitting-set is a vertex hitting set of size 1.…”

Section: Corollariesmentioning

confidence: 61%

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

Heinlein¹,

Ulmer²

2019

Preprint

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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.

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Section: Corollariesmentioning

confidence: 61%

“…As in the vertex version, the set of Hmodels does not have the edge-Erdős-Pósa property for non-planar graphs [10]. But, contrary to the vertex version, the same is true for large (and subcubic) trees and also large ladders [5], which are both planar. Hence only one direction of that equivalence is still true in the edge version.…”

Section: Introductionmentioning

confidence: 91%

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

Heinlein¹,

Ulmer²

2019

Preprint

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“…2 contains a theta graph such that in each at least two subdivided edges meet A; see Figure 4. Again, we find with Lemma 5 kedge-disjoint even A-cycles, which is impossible by (2). □ Let  be the set of strings.…”

Section: Bruhn | 287mentioning

confidence: 78%

“…Now, by identifying the endvertices of the paths in ′ with z, we obtain k edge-disjoint even cycles in G that each pass through z. As ∈ z A we have thus found k edge-disjoint even A-cycles, which is impossible by (2).…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

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Even A‐cycles have the edge‐Erdős–Pósa property

Bruhn

2021

Journal of Graph Theory

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I prove that even A-cycles, cycles of even length that pass through at least one vertex of a prescribed set A, have the edge-Erdős-Pósa property. That is, for any graph and set A, the size of an edge covering of even A-cycles is bounded by a function in the maximal number of edge-disjoint even A-cycles.Recently, it has become ever more clear that there is a significant difference between the ordinary Erdős-Pósa property and its edge-version. Initially it might have seen as if packing and covering were largely the same in the vertex world and in the edge world: as Erdős and Pósa [8] showed every graph either has k vertex-disjoint cycles or a vertex set of size O k k ( log ) meeting all cycles, and similarly, there are always either k edge-disjoint cycles or an edge set of size O k k ( log ) meeting all cycles. The same is true for many other classes of target objects. Say that a class  of graphs has the (ordinary) Erdős-Pósa property (resp., the edge-Erdős-Pósa property) if there is a function f such that for every positive integer k, every graph G either contains k disjoint (resp., edge-disjoint) subgraphs each isomorphic to some graph in  , or G contains a vertex set X (resp., an edge set) of size ≤   X f k ( ) such that G X − is devoid of subgraphs from  . Then not only have cycles both the ordinary and the edge-Erdős-Pósa property but this is also true for even cycles [6,14,3]; for A-cycles 1 [10,11], cycles that each contains at least one vertex from some fixed set A; long cycles [4,13], cycles that have a certain prefixed minimum length; K 4 -subdivisions [1,13]; and many other classes of graphs.

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