The Erdős–Pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces

We prove that for any orientable surface S and any non-negative integer k  , there exists an integer fS(k)fS(k) such that every graph G embeddable in S has either k vertex-disjoint odd cycles or a vertex set A   of cardinality at most fS(k)fS(k) such that G-AG-A is bipartite. Such a property is called the Erdős–Pósa property for odd cycles. We also show its edge version. As Reed [Mangoes and blueberries, Combinatorica 19 (1999) 267–296] pointed out, the Erdős–Pósa property for odd cycles do not hold for all non-orientable surfaces.