The gonality sequence of complete graphs

Abstract

The gonality sequence $({\gamma }_r{)}_{r\ge 1}$ 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 $1\le k\le d-3$ and $0\le h\le k$. 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.