Graphs of gonality three

TL;DR

This paper extends the classification of hyperelliptic graphs to those with divisorial gonality three, providing new constructions and criteria for identifying such graphs under connectivity assumptions.

Contribution

It generalizes Chan's classification to a broader class of graphs with gonality three, including new construction methods and non-gonality conditions.

Findings

Classification extends to gonality three graphs under connectivity assumptions

Provides a construction method for graphs with gonality three

Offers criteria to determine when a graph is not of gonality three

Abstract

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge- and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.