Abstract
A doubly group-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups $\Gamma_1,\Gamma_2$. A cycle in a doubly group-labeled graph is $(\Gamma_1,\Gamma_2)$-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 $(\Gamma_1,\Gamma_2)$-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 $(\Gamma_1,\Gamma_2)$-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.
Highlights
Erdős and Pósa proved in a seminal paper [5] that for every positive integer k, there exists a constant f (k) such that every graph G either contains k pairwise disjoint1 cycles, or a set X ⊆ V (G) of size at most f (k) such that G − X has no cycle
We say that a class G of (Γ1 ⊕ Γ2)-labeled graphs has the half-integral Erdős-Pósa-property (fo−→Gr,(γΓ)1∈, ΓG2)-enitohne-rzero cycles if there exists a function fG(k) such that for every (i) othf e(r−→Ge,isγ)a collection C is contained of k (Γ1, Γ2)-non-zero cycles such in at most two members of C, or that each vertex (ii) there is a set X ⊆ V (G) of size at most fG(k) such that (−→G − X, γ) has no (Γ1, Γ2)-non-zero cycle
We introduce a new notion of Γ-odd clique minors for Γ-labeled graphs, which is useful for attacking problems for non-zero cycles
Summary
Erdős and Pósa proved in a seminal paper [5] that for every positive integer k, there exists a constant f (k) such that every graph G either contains k pairwise disjoint cycles, or a set X ⊆ V (G) of size at most f (k) such that G − X has no cycle. We say that a class G of (Γ1 ⊕ Γ2)-labeled graphs has the half-integral Erdős-Pósa-property (fo−→Gr ,(γΓ)1∈, ΓG2)-enitohne-rzero cycles if there exists a function fG(k) such that for every (i) othf e(r−→Ge ,isγ)a collection C is contained of k (Γ1, Γ2)-non-zero cycles such in at most two members of C, or that each vertex (ii) there is a set X ⊆ V (G) of size at most fG(k) such that (−→G − X, γ) has no (Γ1, Γ2)-non-zero cycle. Theorem 22 provides canonical obstructions to the Erdős-Pósa property for general constrained cycles that are analogous to the Escher-walls described by Reed [16] We derive Theorem 1 and 2 from Theorem 22 in Section 10 and finish with some further applications
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