The Gonality of Queen's Graphs

Abstract

In this paper we study queen's graphs, which encode the moves by a queen on an $n\times m$ chess board, through the lens of chip-firing games. We prove that their gonality is equal to $nm$ minus the independence number of the graph, and give a one-to-one correspondence between maximum independent sets and classes of positive rank divisors achieving gonality. We also prove an identical result for toroidal queen's graphs.