Packing cycles through prescribed vertices under modularity constraints

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.