New Paper on Graph Gonality

The UK Math Lab has submitted a new paper to the arXiv!  The paper, entitled "On the Semigroup of Graph Gonality Sequences", can be found here.

https://arxiv.org/abs/2306.11118

This paper concerns a certain game played on graphs.  (We'll call our players Brill and Noether.)  Brill is given d poker chips to place on the vertices of a graph.  In the example below, d is equal to 2, and Brill may choose to place the chips like this:

 

Noether then "steals" r chips, introducing a debt of r chips.  In the example, r is equal to 1, and Noether might choose to steal a chip like this:

 

Brill's goal is then to eliminate the debt via a sequence of chip-firing moves.  The way this works is that Brill selects a vertex to "fire", which results in that vertex giving 1 chip to each of its neighbors.  If Brill fires the top left vertex, the result is this:

 

This eliminates the debt at the center vertex, but introduces a new debt at the top left.  Brill then fires the bottom left vertex, eliminating all the debt, so Brill wins!

 

If Brill can always win this game no matter how Noether chooses to play, we say that the graph has rth gonality at most d.  In 2020, a group at the SMALL REU asked which sequences can be the gonality sequence of a graph.

https://arxiv.org/abs/2002.07753

In our paper, we consider the set G_r of length-r vectors consisting of the first r terms of the gonality sequence of some graph.  Our key observation is that G_r is a semigroup -- that is, it is closed under addition.  This observation allows us to completely describe G_2, and to begin the study of G_3.

In the last section of the paper, we discuss some questions relating the gonality sequences of graphs to the gonality sequences of algebraic curves.