AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 42 (2008), Pages 141–158 The automorphism group of the toroidal queen’s graph

Abstract

Denote the n × n toroidal queen’s graph by Qt n. We find its automorphism group Aut(Qt n) for each positive integer n, showing that for n ≥ 6, Aut(Qt n) is generated by the translations, the group of the square, the homotheties, and (for odd n) the automorphism (x, y) ↦ → (y + x, y − x). For each n we find the automorphism classes of edges of Qt n,inparticular showing that for n>1, Qt n is edge-transitive if and only if n is prime. We find the number of automorphism classes of regular solutions of the toroidal n-queens problem, generalizing work of Burger, Cockayne, and Mynhardt.