A unified Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

In 1965, Erdős and Pósa proved that there is an (approximate) 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 for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> of integers where such a duality holds for the family of cycles of length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math> modulo <i>z</i>. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.