The Gonality Sequence of Complete Graphs

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 $rplane curve of degree $d$. The same result holds for the corresponding metric graphs.