Half-integral packing of odd cycles through prescribed vertices

The well-known theorem of Erd?s-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 Erd?s-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 Erd?s-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 Erd?s-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 2k 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.