A Unified Erdős–Pósa Theorem for Constrained Cycles

Abstract

A (Γ1, Γ2)-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups Γ1, Γ2. A cycle in such a labeled graph is (Γ1, Γ2)-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 (Γ1, Γ2)-labeled graphs. As an application, we determine all canonical obstructions to the Erdős-Pósa property for (Γ1, Γ2)-non-zero cycles in (Γ1, Γ2)-labeled graphs. The obstructions imply that the halfintegral Erdős-Pósa property always holds for (Γ1, Γ2)-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 Scycles not homologous to zero. Moreover, the (full) Erdős-Pósa property holds for S1-S2-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.

Long cycles through prescribed vertices have the Erdős‐Pósa property

“…Recently, Huynh et al. proved an Erdős‐Pósa type result for cycles in graphs with group labels that includes many previously known results, such as ours, as well as several new ones, such as an Erdős‐Pósa type result for (long) S 1 ‐ S 2 ‐cycles. That much greater generality, however, requires deep tools from minor theory that, in turn, lead to a much worse size of the hitting set.…”

Parity Linkage and the Erdős–Pósa Property of Odd Cycles through Prescribed Vertices in Highly Connected Graphs

“…For S = V (G), these results yield the Erdős-Pósa property for cycles and cycles of length at least , respectively. Several other results about S-cycles are proved in [8].…”

Recent techniques and results on the Erdős–Pósa property

“…If H is a class of graphs, G is a graph and S ⊆ A x (G), then a S-H-subgraph of G is a subgraph of G isomorphic to some member of H and that contain one edge/vertex of S. We are now interested in comparing, for every graph G and every S ⊆ A x (G), the maximum number of S-H-subgraph of G with the minimum number of elements of A x (G) that meet all S-H-subgraphs of G. We refer to these problems by prefixing the guest class with an "S" (like in "S-cycles"). The authors of [HJW16] consider (S 1 , S 2 )-cycles for S 1 , S 2 ⊆ V (G): such cycles must meet both of S and S ′ . A generalization of this type of problem has been introduced in [KM15]: instead of one set S, one considers three subsets S 1 , S 2 , S 3 of V (G) and a (S 1 , S 2 , S 3 )-subgraph is required to intersect at least two sets of S 1 , S 2 and S 3 .…”