Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs

TL;DR

This paper counts the number of nonisomorphic Hamiltonian cycles on square grid graphs, considering all symmetries, and extends previous methods to include various symmetry types for grids up to size 10.

Contribution

It introduces modified enumeration techniques to count nonisomorphic Hamiltonian cycles with different symmetries on square grids up to size 10.

Findings

Counts of nonisomorphic cycles for n<=10

Modified matrix method for symmetry classes

Direct search for 90-degree rotational symmetry

Abstract

The enumeration of Hamiltonian cycles on 2n*2n grids of nodes is a longstanding problem in combinatorics. Previous work has concentrated on counting all cycles. The current work enumerates nonisomorphic cycles -- that is, the number of isomorphism classes (up to all symmetry operations of the square). It is shown that the matrix method used previously can be modified to count cycles with all combinations of reflective and 180-degree rotational symmetry. Cycles with 90-degree rotational symmetry were counted by a direct search, using a modification of Knuth's Dancing Links algorithm. From these counts, the numbers of nonisomorphic cycles were calculated for n<=10.