A note on the threshold numbers of cycles

TL;DR

This paper determines the exact threshold numbers of cycle graphs, showing that the threshold number is 1 for triangles, 2 for quadrilaterals, and 4 for larger cycles, advancing understanding of threshold graph properties.

Contribution

The paper precisely calculates the threshold numbers for all cycle graphs, providing exact values and enhancing the theory of threshold graphs.

Findings

- Threshold number of C_3 is 1. - Threshold number of C_4 is 2. - Threshold number of C_n for n ≥ 5 is 4.

The results clarify the relationship between cycle length and threshold number.

Abstract

A graph $G=(V,E)$ is said to be a \textit{$k$-threshold graph} with \textit{thresholds} $\theta_1<\theta_2<...<\theta_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ 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 \textit{threshold number} of $G$, denoted by $\Theta(G)$, 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 \[ \Theta(C_n)=\begin{cases} 1 & if\ n=3, 2 & if\ n=4, 4 & if\ n\ge 5, \end{cases} \] where $C_n$ is the cycle with $n$ vertices.