O-joung Kwon (권오정), Erdős-Pósa property of A-paths in unoriented group-labelled graphs

A family $\mathcal{F}$ of graphs is said to satisfy the Erdős-Pósa property if there exists a function $f$ such that for every positive integer $k$, every graph $G$ contains either $k$ (vertex-)disjoint subgraphs in $\mathcal{F}$ or a set of at most $f(k)$ vertices intersecting every subgraph of $G$ in $\mathcal{F}$. 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 $\mathrm{\Gamma }$ and for every subset $\mathrm{\Lambda }$ of $\mathrm{\Gamma }$, the family of $\mathrm{\Gamma }$-labelled $A$-paths whose lengths are in $\mathrm{\Lambda }$ satisfies the half-integral relaxation of the Erdős-Pósa property. Moreover, we give a characterization of such $\mathrm{\Gamma }$ and $\mathrm{\Lambda }\subseteq \mathrm{\Gamma }$ for which the same family of $A$-paths satisfies the full Erdős-Pósa property. This is joint work with Youngho Yoo.