The chip-firing game is a simple game with only one move. Starting with
a collection of poker chips on the vertices of a graph, one is allowed
to move them around the graph using what is known as the "chip-firing
move". A version of this game can be found here. Despite its simplicity, this game has deep connections to dynamics, number theory, and algebraic geometry.
The gonality of a graph is the smallest number of chips required so that the chip-firing game is always winnable. In this project, we will attempt to discern the gonality of a graph product from the gonality of each of its factors. A natural conjecture is that
gon(GxH) = min{|V(G)|gon(H),|V(H)|gong(G)}.
This project will focus heavily on examples, seeking either a counterexample to the above conjecture, or to verify it in a number of cases.
The gonality of graphs has attracted recent interest, due to its connection to the geometry of algebraic curves. Graph products are of particular interest because of their relation to the geometry of complete intersection curves.
The gonality of a graph is the smallest number of chips required so that the chip-firing game is always winnable. In this project, we will attempt to discern the gonality of a graph product from the gonality of each of its factors. A natural conjecture is that
gon(GxH) = min{|V(G)|gon(H),|V(H)|gong(G)}.
This project will focus heavily on examples, seeking either a counterexample to the above conjecture, or to verify it in a number of cases.
The gonality of graphs has attracted recent interest, due to its connection to the geometry of algebraic curves. Graph products are of particular interest because of their relation to the geometry of complete intersection curves.
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