A unified Erd\H{o}s-P\'osa theorem for constrained cycles

A doubly group-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$. A cycle in a doubly group-labeled graph is $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)$-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 doubly group-labeled graphs. As an application, we determine all canonical obstructions to the Erd\H{o}s-P'osa property for $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)$-non-zero cycles in doubly group-labeled graphs. The obstructions imply that the half-integral Erd\H{o}s-P'osa property always holds for $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)$-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 Erd\H{o}s-P'osa property for cycles and $S$-cycles and the half-integral Erd\H{o}s-P'osa 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 Erd\H{o}s-P'osa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and $S$-cycles not homologous to zero. Moreover, the (full) Erd\H{o}s-P'osa 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 Erd\H{o}s-P'osa property for cycles not homologous to zero and for odd $S$-cycles.