On the Gonality of Ferrers Rook Graphs

TL;DR

This paper investigates the gonality of Ferrers rook graphs, proposing a conjectural formula, proving it for specific families and small cases, advancing understanding of their algebraic and combinatorial properties.

Contribution

It introduces a conjectural formula for the gonality of Ferrers rook graphs and proves it for certain infinite families and small diagrams, extending previous knowledge.

Findings

Conjectural formula for gonality proposed

Proved for specific infinite families of diagrams

Confirmed for all diagrams with up to 8 dots

Abstract

A Ferrers rook graph is a graph whose vertices correspond to the dots in a Ferrers diagram, and where two vertices are adjacent if they are in the same row or the same column. We propose a conjectural formula for the gonality of Ferrers rook graphs, and prove this conjecture for a few infinite families of Ferrers diagrams. We also prove the conjecture for all Ferrers diagrams $F$ with $|F| \leq 8$.