This talk follows on from the recent talk of Pascal Gollin in this seminar series, but will aim to be accessible for newcomers.
Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. By relaxing `packing’ to `half-integral packing’, Reed obtained an analogous result for odd cycles, and gave a structural characterisation of when the (integral) packing version fails.
We prove some far-reaching generalisations of these theorems. First, we show that if the edges of a graph are labelled by finitely many abelian groups, then the cycles whose values avoid a fixed finite set for each abelian group satisfy the half-integral Erdős-Pósa property. Similarly to Reed, we give a structural characterisation for the failure of the integral Erdős-Pósa property in this setting. This allows us to deduce the full Erdős-Pósa property for many natural classes of cycles.
We will look at applications of these results to graphs embedded on surfaces, and also discuss some possibilities and obstacles for extending these results.
This is joint work with Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum, and Youngho Yoo.
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