Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs
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 180degree rotational symmetry. Cycles with 90degree 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.0545
 Bibcode:
 2014arXiv1402.0545W
 Keywords:

 Mathematics  Combinatorics;
 05C45