Abstract
In this paper we provide the first systematic treatment of Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve defined in terms of chip-firing. We prove an upper bound on the gonality of the Cartesian product of any two graphs, and determine 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. We also extend some of our results to metric graphs.
Highlights
In [7], Baker and Norine introduced a theory of divisors on finite graphs in parallel to divisor theory on algebraic curves
If G = (V, E) is a connected multigraph, one treats G as a discrete analog of an algebraic curve of genus g(G), where g(G) = |E| − |V | + 1. This program was extended to metric graphs in [15] and [21], and has been used to study algebraic curves through combinatorial means
We study the gonality of the Cartesian product G H of two graphs G and H
Summary
In [7], Baker and Norine introduced a theory of divisors on finite graphs in parallel to divisor theory on algebraic curves. We study the gonality of the Cartesian product G H of two graphs G and H. For many naturally occurring examples of G and H where gon(G H) is known, the inequality is an equality This leads us to pose the following question. 7 and 8 we will see that the gap between gonality and expected gonality can be arbitrarily large, both when considering simple and non-simple graphs.
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