### Abstract

The well-known theorem of Erdos-Pósa says that either a graph G has k disjoint cycles or there is a vertex set X of order at most f(k) for some function f such that G/X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we present a common generalization of the following Erdos-Pósa properties: The Erdos-Pósa property for cycles of length divisible by a fixed integer p (Thomassen, 1988 [19]).The Erdos-Pósa property for S-cycles, i.e., cycles which contain a vertex of a prescribed vertex set S (Kakimura, Kawarabayashi, and Marx, 2011 [10] and Pontecorvi and Wollan, 2010 [13]). Namely, given integers k,p, and a vertex set S (whose size may not depend on k and p), we prove that either a graph G has k disjoint S-cycles, each of which has length divisible by p, or G has a vertex set X of order at most f(k,p) such that G/X has no such a cycle.

Original language | English |
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Pages (from-to) | 97-110 |

Number of pages | 14 |

Journal | Advances in Applied Mathematics |

Volume | 49 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Aug 1 |

Externally published | Yes |

### Keywords

- Disjoint cycles
- Erdos-Pósa property
- Even cycles
- Feedback vertex sets

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*Advances in Applied Mathematics*,

*49*(2), 97-110. https://doi.org/10.1016/j.aam.2012.03.002