Discrete and metric divisorial gonality can be different

Abstract

This paper compares the divisorial gonality of a finite graph $G$ to the divisorial gonality of the associated metric graph $\hat{I}“(G,1)$ with unit lengths. We show that $\text{dgon}(\hat{I}“(G,1))$ 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.

15 pages, 4 figures. Changes: improved Lemma 4.4, added Proposition 5.3, changed open question 2