On the gonality of Cartesian products of graphs

TL;DR

This paper investigates the divisorial gonality of Cartesian product graphs, establishing an upper bound, identifying cases of equality, and confirming Baker's gonality conjecture for these graphs.

Contribution

It provides a new upper bound on the gonality of Cartesian product graphs and verifies Baker's conjecture in this context.

Findings

Upper bound on gonality for Cartesian product graphs

Instances where the bound is tight, including rook's graphs

Confirmation of Baker's gonality conjecture for these graphs

Abstract

In this paper we study Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve. We present an upper bound on the gonality of the Cartesian product of any two graphs, and provide instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$. We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound.