A note on the threshold numbers of cycles

A graph $G=(V,E)$ is said to be a $k\mathit{\text{-threshold}}\mathit{\text{graph}}$ with $\mathit{\text{thresholds}}$ ${\theta }_1<{\theta }_2<\mathrm{.}\mathrm{.}\mathrm{.}<{\theta }_k$ if there is a map $r:V⟶\mathbb{R}$ such that $uv\in E$ if and only if ${\theta }_i\le r(u)+r(v)$ holds for an odd number of $i\in [k]$. The $\mathit{\text{threshold}}\mathit{\text{number}}$ of $G$, denoted by $\mathrm{\Theta }(G)$, 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 $$\mathrm{\Theta }(C_n)=\{\begin{array}{llll}1& ifn=3,2& ifn=4,4& ifn\ge 5,\end{array}$$ where $C_n$ is the cycle with $n$ vertices.