Packing cycles in undirected group-labelled graphs

Abstract

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,{\hat{I}}^3)$ where ${\hat{I}}^3$ assigns to each edge of an undirected graph $G$ an element of an abelian group $\hat{I}“$. As a consequence, we prove that $\hat{I}“$-nonzero cycles (cycles whose edges sum to a non-identity element of $\hat{I}“$) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that, if $\hat{I}“$ has no element of order two, then $\hat{I}“$-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if $m$ is an odd prime power, then cycles of length $\ell \mathrm{mod}m$ satisfy the Erdős-Pósa property for all integers $\ell$. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs $(\ell ,m)$.