The gonality sequence of complete graphs

TL;DR

This paper determines the gonality sequence of complete graphs and their metric versions, showing it matches that of a smooth plane curve of degree d, thus linking graph gonality to algebraic geometry.

Contribution

It explicitly computes the gonality sequence for complete graphs and metric graphs, revealing a precise formula and its geometric significance.

Findings

Gonality sequence of K_d is given by a specific formula involving k and h.

The sequence matches that of a smooth plane curve of degree d.

Results extend to metric graphs associated with K_d.

Abstract

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.