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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2009.11266 |
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| _version_ | 1866913401959088128 |
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| author | Thomas, Robin Yoo, Youngho |
| author_facet | Thomas, Robin Yoo, Youngho |
| contents | We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,γ)$ where $γ$ assigns to each edge of an undirected graph $G$ an element of an abelian group $Γ$. As a consequence, we prove that $Γ$-nonzero cycles (cycles whose edges sum to a non-identity element of $Γ$) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that, if $Γ$ has no element of order two, then $Γ$-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if $m$ is an odd prime power, then cycles of length $\ell \mod m$ 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 $(\ell,m)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_11266 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Packing cycles in undirected group-labelled graphs Thomas, Robin Yoo, Youngho Combinatorics We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,γ)$ where $γ$ assigns to each edge of an undirected graph $G$ an element of an abelian group $Γ$. As a consequence, we prove that $Γ$-nonzero cycles (cycles whose edges sum to a non-identity element of $Γ$) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that, if $Γ$ has no element of order two, then $Γ$-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if $m$ is an odd prime power, then cycles of length $\ell \mod m$ 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 $(\ell,m)$. |
| title | Packing cycles in undirected group-labelled graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2009.11266 |