Looking for the Hardest Hamiltonian Cycle Problem Instances

We use two evolutionary algorithms to make hard instances of the Hamiltonian cycle problem. Hardness, or fitness, is defined as the number of recursions required by Vandegriend-Culberson, the best known exact backtracking algorithm for the problem. The hardest instances, all non-Hamiltonian, display a high degree of regularity and scalability across graph sizes. These graphs are found multiple times through independent runs and in both algorithms, suggestion the search space might contain monotonic paths to the global maximum. The one-bit neighbourhoods of these instances are also analyzed, and show that these hardest instances are separated from the easiest problem instances by just one bit of information. For Hamiltonian-bound graphs, the hardest instances are less uniform and substantially easier than their non-Hamiltonian counterparts. Reasons for these less-conclusive results are presented and discussed.