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Pascal Gollin, A unified Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups

February 8 Tuesday @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)

Speaker

Pascal Gollin
IBS Discrete Mathematics Group
https://dimag.ibs.re.kr/home/gollin/

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. We therefore say that cycles satisfy the Erdős-Pósa property. However, while odd cycles do not satisfy the Erdős-Pósa property, Reed proved in 1999 an analogue by relaxing packing to half-integral packing, where each vertex is allowed to be contained in at most two such cycles. Moreover, he gave a structural characterisation for when the Erdős-Pósa property for odd cycles fails.

We prove a far-reaching generalisation of the theorem of Reed; 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, and we similarly give a structural characterisation for the failure of the Erdős-Pósa property.

A multitude of natural properties of cycles can be encoded in this setting. For example, we show that the cycles of length $\ell$ modulo $m$ satisfy the half-integral Erdős-Pósa property, and we characterise for which values of $\ell$ and $m$ these cycles satisfy the Erdős-Pósa property.

This is joint work with Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum, and Youngho Yoo.

Details

Date:
February 8 Tuesday
Time:
4:30 PM - 5:30 PM KST
Event Category:
Event Tags:

Venue

Room B232
IBS (기초과학연구원)

Organizer

Sang-il Oum (엄상일)
View Organizer Website
IBS 이산수학그룹 Discrete Mathematics Group
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209
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