The Gonality of Rook Graphs

TL;DR

This paper determines the gonality of two-dimensional rook graphs, showing it equals the product of the smaller complete graph size minus one and the larger size, and explores related invariants.

Contribution

It proves the gonality formula for rook graphs and computes their 2 and 3 gonalities, introducing the scramble number as a new lower bound.

Findings

Gonality of rook graphs is (n-1)m where n and m are sizes of the complete graphs.

Computed the 2 and 3 gonalities of rook graphs.

Introduced the scramble number as a new invariant and lower bound.

Abstract

Two dimensional rook graphs are the Cartesian product of two complete graphs. In this paper we prove that the gonality of these graphs is the expected value of $(n-1)m$ where $n$ is the size of the smaller complete graph and $m$ is the size of the larger. furthermore we compute the 2 and 3 gonalities of these graphs. We also explore the scramble number of these graphs, which is a new graph invariant and a lower bound on the gonality.