The Erdős–Pósa Property for Long Circuits

Meierling, Dirk; Rautenbach, Dieter; Sasse, Thomas

doi:10.1002/jgt.21769

Cited by 6 publications

(8 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When revising the current article, we learned of the following recent improvement to Lemma 2.2. Lemma 4.1 (Meierling, Rautenbach and Sasse [6]). If a graph G does not contain two vertex-disjoint long cycles, then it contains a set of at most 5 3 + 29 2 vertices that meets all the long cycles.…”

Section: Discussionmentioning

confidence: 99%

A Tighter Erdős-Pósa Function for Long Cycles

Fiorini

Herinckx

2013

J. Graph Theory

View full text Add to dashboard Cite

We prove that there exists a bivariate function f with f(k,ℓ)=O(ℓ·klogk) such that for every natural k and ℓ, every graph G has at least k vertex‐disjoint cycles of length at least ℓ or a set of at most f(k,ℓ) vertices that meets all cycles of length at least ℓ. This improves a result by Birmelé et al. (Combinatorica, 27 (2007), 135–145), who proved the same result with f(k,ℓ)=Θ(ℓ·k2).

show abstract

Section: Discussionmentioning

confidence: 99%

A Tighter Erdős-Pósa Function for Long Cycles

Fiorini

Herinckx

2013

J. Graph Theory

View full text Add to dashboard Cite

show abstract

“…Strikingly, this case turns out to be the longest and most involved part in the argumentation of [1]. While the result was recently improved by Meierling et al [16], the proof still takes a substantial effort. Moreover, the nonconstructive nature of the proof makes it difficult to extract an algorithm.…”

Section: Theorem 6 (Simonovits [22]) Every Cubic Multigraph With At mentioning

confidence: 80%

“…While the result was recently improved by Meierling et al. , the proof still takes a substantial effort. Moreover, the nonconstructive nature of the proof makes it difficult to extract an algorithm.…”

Section: Preliminaries and Short Discussionmentioning

confidence: 98%

Long cycles through prescribed vertices have the Erdős‐Pósa property

Bruhn

Joos

Schaudt

2017

Journal of Graph Theory

View full text Add to dashboard Cite

We prove that for every graph, any vertex subset S, and given integers k,ℓ: there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size O(ℓ·klogk) that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in O(klogk·s2·false(f(ℓ)·n+mfalse)) time, where s denotes the cardinality of S.

show abstract

“…For larger l, Birmelé, Bondy, and Reed [3] proved that the optimal function satisfies f (l, 2) ≤ 2l + 3. This was recently improved by Meierling, Rautenbach and Sasse [16] to f (l, 2) ≤ 5l/3 + 29/2.…”

Section: Introductionmentioning

confidence: 97%

A tight Erdős–Pósa function for long cycles

Mousset

Noever

Škorić

et al. 2017

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

A classic result of Erdős and Pósa states that any graph either contains k vertexdisjoint cycles or can be made acyclic by deleting at most O(k log k) vertices. Birmelé, Bondy, and Reed (2007) raised the following more general question: given numbers l and k, what is the optimal function f (l, k) such that every graph G either contains k vertex-disjoint cycles of length at least l or contains a set X of f (l, k) vertices that meets all cycles of length at least l? In this paper, we answer that question by proving that f (l, k) = Θ(kl + k log k). As a corollary, the tree-width of any graph G that does not contain k vertexdisjoint cycles of length at least l is of order O(kl + k log k). This is also optimal up to constant factors and answers another question of Birmelé, Bondy, and Reed (2007).

show abstract

The Erdős–Pósa Property for Long Circuits

Cited by 6 publications

References 4 publications

A Tighter Erdős-Pósa Function for Long Cycles

A Tighter Erdős-Pósa Function for Long Cycles

Long cycles through prescribed vertices have the Erdős‐Pósa property

A tight Erdős–Pósa function for long cycles

Contact Info

Product

Resources

About