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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 by both evolutionary algorithms, suggesting the search space might contain monotonic paths towards the global maximum. For Hamiltonian-bound evolution, some hard graphs were found, but convergence is much less consistent. In this extended paper, we survey the neighbourhoods of both the hardest yes- and no-instances produced by the evolutionary algorithms. Results show that the hardest no-instance resides on top of a steep cliff, while the hardest yes-instance turns out to be part of a plateau of 27 equally hard instances. While definitive answers are far away, the results provide a lot of insight in the Hamiltonian cycle problem’s state space.

Even though the Hamiltonian cycle problem is NPcomplete, many of its problem instances aren't. In fact, almost all the hard instances reside in one area: near the Komlós-Szemerédi bound, ofedges, where randomly generated graphs have an approximate 50% chance of being Hamiltonian. If the number of edges is either much higher or much lower, the problem is not hard -most backtracking algorithms decide such instances in (near) polynomial time. Recently however, targeted search efforts have identified very hard Hamiltonian cycle problem instances very far away from the Komlós-Szemerédi bound. In that study, the used backtracking algorithm was Vandegriend-Culberson's, which was supposedly the most efficient of all Hamiltonian backtracking algorithms.In this paper, we make a unified large scale quantitative comparison for the best known backtracking algorithms described between 1877 and 2016. We confirm the suspicion that the Komlós-Szemerédi bound is a hard area for all backtracking algorithms, but also that Vandegriend-Culberson is indeed the most efficient algorithm, when expressed in consumed computing time. When measured in recursive effectiveness however, the algorithm by Frank Rubin, almost half a century old, performs best. In a more general algorithmic assessment, we conjecture that edge pruning and non-Hamiltonicity checks might be largely responsible for these recursive savings. When expressed in system time however, denser problem instances require much more time per recursion. This is most likely due to the costliness of the extra search pruning procedures, which are relatively elaborate. We supply large amounts of experimental data, and a unified single-program implementation for all six algorithms. All data and algorithmic source code is made public for further use by our colleagues.