A note on the threshold numbers of cycles

A graph $G=(V,E)$ is said to be a k-threshold graph with thresholds θ₁<θ₂<...<θₖ if there is a map r: V R such that uv∈ E if and only if θᵢ≤ r(u)+r(v) holds for an odd number of i∈ [k]. The 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 \[ Θ(C_n)={cases} 1 & if\ n=3, 2 & if\ n=4, 4 & if\ n≥ 5, {cases} \] where Cₙ is the cycle with n vertices.