## Abstract

The well-known theorem of Erdős-Pósa says that a graph *G* has either *k* disjoint cycles or 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 discuss packing disjoint *S*-cycles, i.e., cycles that are required to go through a set *S* of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erdős-Pósa-type result holds. We further try to generalize this result to packing *S*-cycles of odd length. In contrast to packing *S*-cycles, the Erdős-Pósa-type result does not hold for packing odd *S*-cycles. We then relax packing odd *S*-cycles to half-integral packing, and show the Erdős-Pósa-type result for the half-integral packing of odd *S*-cycles, which is a generalization of Reed (1999) when *S*=*V*. That is, we show that given an integer *k* and a vertex set *S*, a graph *G* has either 2*k* odd *S*-cycles so that each vertex is in at most two of these cycles, or a vertex set *X* of order at most *f*(*k*) (for some function *f*) such that *G\X* has no odd *S*-cycle.

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## Additional information

A preliminary version of this paper appears as a part of “Erdős-Pósa property and its algorithmic applications — parity constraints, subset feedback set, and subset packing”[11] in *Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012)*.

Partly supported by Grant-in-Aid for Scientific Research and JST, ERATO, Kawarabayashi Large Graph Project.

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Kakimura, N., Kawarabayashi, KI. Half-integral packing of odd cycles through prescribed vertices.
*Combinatorica* **33, **549–572 (2013). https://doi.org/10.1007/s00493-013-2865-6

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### Mathematics Subject Classification (2000)

- 68R10
- 05C38
- 05C70