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Using an improved backtrack algorithm with sophisticated pruning techniques,we revise previous observations correlating a high frequency of hard to solveHamiltonian Cycle instances with the Gn,m phase transition betweenHamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500vertices are easily solved. When we artificially restrict the degree sequencewith a bounded maximum degree, although there is some increase in difficulty,the frequency of hard graphs is still low. When we consider more regular graphsbased on a generalization of knight's tours, we observe frequent instances ofreally hard graphs, but on these the average degree is bounded by a constant.We design a set of graphs with a feature our algorithm is unable to detect andso are very hard for our algorithm, but in these we can vary the average degreefrom O(1) to O(n). We have so far found no class of graphs correlated with theGn,m phase transition which asymptotically produces a high frequency of hardinstances.

The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem