A note on the threshold numbers of cycles
Runze Wang
TL;DR
This paper determines the exact threshold numbers for cycle graphs in the multithreshold framework. It defines $k$-threshold representations with thresholds $\theta_i$ and rank function $r$, and derives the exact values $\Theta(C_3)=1$, $\Theta(C_4)=2$, and $\Theta(C_n)=4$ for all $n\ge 5$. The main approach combines explicit constructions to realize a 4-threshold representation and parity-based contradictions to eliminate smaller $k$, including a $4$-threshold assignment with $r(v_i)=(-1)^{i-1} i$ for $i\le n-1$ and $r(v_n)=(-1)^{n-1}(n-0.5)$ and carefully chosen thresholds. The results complete the landscape of threshold numbers for cycles and connect with known bounds for paths and multipartite graphs, offering both a concrete representation and a nonexistence proof for $k\in\{2,3\}$ when $n\ge 5$.
Abstract
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.