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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.13955 |
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| _version_ | 1866916755637534720 |
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| author | Wang, Runze |
| author_facet | Wang, Runze |
| contents | A graph $G=(V,E)$ is said to be a \textit{$k$-threshold graph} with \textit{thresholds} $θ_1<θ_2<...<θ_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ such that $uv\in E$ if and only if $θ_i\le r(u)+r(v)$ holds for an odd number of $i\in [k]$. The \textit{threshold number} of $G$, denoted by $Θ(G)$, is the smallest positive integer $k$ such that $G$ is a $k$-threshold graph. In this paper, we determine the exact threshold numbers of cycles by proving
\[ Θ(C_n)=\begin{cases}
1 & if\ n=3,
2 & if\ n=4,
4 & if\ n\ge 5,
\end{cases}
\]
where $C_n$ is the cycle with $n$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the threshold numbers of cycles Wang, Runze Combinatorics A graph $G=(V,E)$ is said to be a \textit{$k$-threshold graph} with \textit{thresholds} $θ_1<θ_2<...<θ_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ such that $uv\in E$ if and only if $θ_i\le r(u)+r(v)$ holds for an odd number of $i\in [k]$. The \textit{threshold number} of $G$, denoted by $Θ(G)$, is the smallest positive integer $k$ such that $G$ is a $k$-threshold graph. In this paper, we determine the exact threshold numbers of cycles by proving \[ Θ(C_n)=\begin{cases} 1 & if\ n=3, 2 & if\ n=4, 4 & if\ n\ge 5, \end{cases} \] where $C_n$ is the cycle with $n$ vertices. |
| title | A note on the threshold numbers of cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.13955 |