The gonality sequence of complete graphs

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The gonality sequence (γᵣ)ᵣ≥₁ of a finite graph / metric graph / algebraic curve comprises the minimal degrees γᵣ of linear systems of rank r. For the complete graph Kd, we show that γᵣ = kd - h if r<g=(d-1)(d-2)/2, where k and h are the uniquely determined integers such that r = k(k+3)/2 - h with 1≤ k≤ d-3 and 0 ≤ h ≤ k. This shows that the graph Kd has the gonality sequence of a smooth plane curve of degree d. The same result holds for the corresponding metric graphs.