## Abstract

The rth gonality of a graph is the smallest degree of a divisor on the graph with rank r. The gonality sequence of a graph is a tropical analogue of the gonality sequence of an algebraic curve. We show that the set of truncated gonality sequences of graphs forms a semigroup under addition. Using this, we study which triples (x, y, z) can be the first three terms of a graph gonality sequence. We show that nearly every such triple with (Formula present) is the first three terms of a graph gonality sequence, and also exhibit triples where the ratio (Formula present) is an arbitrary rational number between 1 and 3. In the final section, we study algebraic curves whose rth and (r + 1) st gonality differ by 1, and posit several questions about graphs with this property.

Original language | English |
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Pages (from-to) | 343-361 |

Number of pages | 19 |

Journal | Australasian Journal of Combinatorics |

Volume | 88 |

Issue number | 3 |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© The author(s).

### Funding

This research was conducted as a project with the University of Kentucky Math Lab, supported by NSF DMS-2054135. We would like to thank the anonymous referees for their thoughtful and helpful comments.

Funders | Funder number |
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University of Kentucky Math Lab | |

National Science Foundation Arctic Social Science Program | DMS-2054135 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics