TY - JOUR

T1 - A tight Erdős–Pósa function for long cycles

AU - Mousset, F.

AU - Noever, A.

AU - Škorić, N.

AU - Weissenberger, F.

N1 - Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2017/7

Y1 - 2017/7

N2 - A classic result of Erdős and Pósa states that any graph either contains k vertex-disjoint cycles or can be made acyclic by deleting at most O(klogk) 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+klogk). As a corollary, the tree-width of any graph G that does not contain k vertex-disjoint cycles of length at least l is of order O(kl+klogk). This is also optimal up to constant factors and answers another question of Birmelé, Bondy, and Reed (2007).

AB - A classic result of Erdős and Pósa states that any graph either contains k vertex-disjoint cycles or can be made acyclic by deleting at most O(klogk) 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+klogk). As a corollary, the tree-width of any graph G that does not contain k vertex-disjoint cycles of length at least l is of order O(kl+klogk). This is also optimal up to constant factors and answers another question of Birmelé, Bondy, and Reed (2007).

KW - Covering vs packing

KW - Erdős–Pósa property

KW - Graph theory

KW - Long cycles

UR - http://www.scopus.com/inward/record.url?scp=85012880026&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2017.01.004

DO - 10.1016/j.jctb.2017.01.004

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AN - SCOPUS:85012880026

SN - 0095-8956

VL - 125

SP - 21

EP - 32

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -