Treewidth is a lower bound on graph gonality

TL;DR

This paper establishes that the treewidth of a graph provides a fundamental lower bound for its gonality, extending the relationship to metric graphs and other gonality notions, with exact matches for specific graph classes.

Contribution

It proves the treewidth lower bound for graph gonality and extends this result to metric graphs and related gonality concepts, providing a unified theoretical framework.

Findings

Treewidth bounds gonality for finite connected graphs.

Equality of gonality and treewidth for grid and complete multipartite graphs.

Treewidth lower bound extends to metric graphs and other gonality notions.

Abstract

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for \emph{metric graphs} by constructing for any positive rank divisor on a metric graph $\Gamma$ a positive rank divisor of the same degree on a subdivision of the underlying graph. Finally, we show that the treewidth lower bound also holds for a related notion of gonality defined by Caporaso and for stable gonality as introduced by Cornelissen et al.