Aharoni's rainbow cycle conjecture holds up to an additive constant
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915062065659904 |
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| author | Hompe, Patrick Huynh, Tony |
| author_facet | Hompe, Patrick Huynh, Tony |
| contents | In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$.
In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $α_r \in O(r^5 \log^2 r)$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + α_r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05697 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Aharoni's rainbow cycle conjecture holds up to an additive constant Hompe, Patrick Huynh, Tony Combinatorics Discrete Mathematics 05C38, 05C20, 05C15 In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $α_r \in O(r^5 \log^2 r)$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + α_r$. |
| title | Aharoni's rainbow cycle conjecture holds up to an additive constant |
| topic | Combinatorics Discrete Mathematics 05C38, 05C20, 05C15 |
| url | https://arxiv.org/abs/2212.05697 |