Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example, cycles of length at least $\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times and cycles contained in given ${\mathbb{Z}}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.

中文翻译：

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多个阿贝尔群标记图中循环的统一半积分 Erdős-Pósa 定理

Erdős 和 Pósa 在 1965 年证明，循环堆积的最大尺寸和命中所有循环的顶点集的最小尺寸之间存在对偶性。如果我们限制奇数循环，这种对偶性就不成立。然而，在 1999 年，Reed 通过将堆积松弛为半积分堆积证明了奇循环的类似物。我们证明了里德定理的深远概括；如果图的边由有限多个交换群标记，则循环的半积分堆积的最大大小（其值避免每个交换群的固定有限集）与顶点集的最小大小之间存在对偶性击中所有此类循环。循环的多种自然属性可以在此设置中编码，例如，长度至少为 $\ell$，周期长度 $p$模数 $q$，至少与一组规定的顶点相交的循环 $t$给定的时间和周期${\mathbb{Z}}_2$-嵌入在固定表面上的图中的同源类。我们的主要结果使我们能够证明满足有限多个此类属性的固定集合的循环的对偶定理。