Discrete and metric divisorial gonality can be different

TL;DR

This paper investigates the relationship between divisorial gonality of finite graphs and their associated metric graphs, revealing that they can differ and providing counterexamples to a previous conjecture.

Contribution

It establishes that the divisorial gonality of a metric graph equals the minimal gonality among all regular subdivisions of the original graph, disproving Baker's conjecture.

Findings

Divisorial gonality of metric graphs can be strictly smaller than that of the original graph.

The paper characterizes the gonality of associated metric graphs as a minimal subdivision gonality.

Counterexamples to Baker's conjecture are provided.

Abstract

This paper compares the divisorial gonality of a finite graph $G$ to the divisorial gonality of the associated metric graph $\Gamma(G,\mathbb{1})$ with unit lengths. We show that $\text{dgon}(\Gamma(G,\mathbb{1}))$ is equal to the minimal divisorial gonality of all regular subdivisions of $G$, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of $G$. This settles a conjecture of M. Baker in the negative.