Erdős-Pósa property of $A$-paths in unoriented group-labelled graphs
O-joung Kwon, Youngho Yoo
TL;DR
The paper addresses the Erdős-Pósa property for $A$-paths in unoriented group-labelled graphs and identifies the obstructions to both the half-integral and full versions of the property. It proves that for any finite abelian group $Γ$ and subset $Λ\subseteq Γ$, the family of $Λ$-allowable $A$-paths satisfies the half-integral Erdős-Pósa property, and it provides a precise Erdős-Pósa condition on $Λ$ that characterizes when the full property holds. Central to the approach is a structure theorem that reduces non-packing instances to special $igl(A,kigr)$-ribboned walls built from balanced walls and handlebars, together with careful use of walls, handlebars, and tangles. The results extend and unify prior work on cycles and labeled paths, offering a constructive, computable bound and a framework for broader endpoint-constraint families, with potential extensions to certain infinite-group settings. Overall, the work advances the understanding of how label-based obstructions control packing and transversal properties in group-labelled graphs, with implications for related extremal and topological graph theory questions.
Abstract
We characterize the obstructions to the Erdős-Pósa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $Γ$ and for every subset $Λ$ of $Γ$, the family of $Γ$-labelled $A$-paths whose lengths are in $Λ$ satisfies the half-integral Erdős-Pósa property. Moreover, we give a characterization of such $Γ$ and $Λ\subseteqΓ$ for which the same family of $A$-paths satisfies the full Erdős-Pósa property.