The gonality sequence $(\gamma_r)_{ r\geq1}$ of a finite graph / metric graph / algebraic curve comprises the minimal degrees $\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\gamma_r = kd - h$ if $r<g=\frac{ (d-1)(d-2)}{ 2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \frac{ k(k+3)}{ 2} - h$ with $1\leq k\leq d-3$ and $0 \leq h \leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs.
Categories: math.CO math.AG