Crossing number of graphs and $\mathsf{ΔY}$-move
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916128302825472 |
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| author | Huh, Youngsik Nikkuni, Ryo |
| author_facet | Huh, Youngsik Nikkuni, Ryo |
| contents | The crossing number of a graph is the minimum number of double points over all generic immersions of the graph into the plane. In this paper we investigate the behavior of crossing number under a graph transformation, called $\mathsf{ΔY}$-move, on the complete graph $K_n$. Concretely it is shown that for any $k\in \mathbb{N}$, there exist a natural number $n$ and a sequence of $\mathsf{ΔY}$-moves $K_n\rightarrow G^{(1)}\rightarrow \cdots \rightarrow G^{(k)}$ which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crossing number of graphs and $\mathsf{ΔY}$-move Huh, Youngsik Nikkuni, Ryo Combinatorics Geometric Topology 05C62, 57M15, 05C10 The crossing number of a graph is the minimum number of double points over all generic immersions of the graph into the plane. In this paper we investigate the behavior of crossing number under a graph transformation, called $\mathsf{ΔY}$-move, on the complete graph $K_n$. Concretely it is shown that for any $k\in \mathbb{N}$, there exist a natural number $n$ and a sequence of $\mathsf{ΔY}$-moves $K_n\rightarrow G^{(1)}\rightarrow \cdots \rightarrow G^{(k)}$ which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small $n$. |
| title | Crossing number of graphs and $\mathsf{ΔY}$-move |
| topic | Combinatorics Geometric Topology 05C62, 57M15, 05C10 |
| url | https://arxiv.org/abs/2402.10633 |