A Unified Erdős–Pósa Theorem for Constrained Cycles

A (Γ 1 ,Γ 2 )-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups Γ 1 ,Γ 2 . A cycle in such a labeled graph is (Γ 1 ,Γ 2 )-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to (Γ 1 ,Γ 2 )-labeled graphs. As an application, we determine all canonical obstructions to the Erdős–Pósa property for (Γ 1 ,Γ 2 )-non-zero cycles in (Γ 1 ,Γ 2 )-labeled graphs. The obstructions imply that the half-integral Erdős–Pósa property always holds for (Γ 1 ,Γ 2 )-non-zero cycles. Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erdős–Pósa property for cycles and S-cycles and the half-integral Erdős–Pósa property for odd cycles and odd S-cycles. Furthermore, we recover Reed's Escher-wall Theorem. We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erdős–Pósa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and S-cycles not homologous to zero. Moreover, the (full) Erdős–Pósa property holds for S 1 -S 2 -cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erdős–Pósa property for cycles not homologous to zero and for odd S-cycles.