Systematic generation of very hard cases for graph 3-colorability

We present a simple generation procedure which turns out to be an effective source of very hard cases for graph 3-colorability. The graphs distributed according to this generation procedure are much denser in very hard cases than previously reported for the same problem size. The coloring cost for these instances is also orders of magnitude bigger. This ability is issued from the fact that the procedure favors-inside the class of graphs with given connectivity and free of 4-cliques-the generation of graphs with relatively few paths of length three (that we call 3-paths). There is a critical value of the ratio between the number of 3-paths and the number of edges, independent of the number of nodes, which separates the graphs having the same connectivity in two regions: one contains almost all graphs free of 4-cliques while the other contains almost no such graphs. The generated very hard cases are near this phase transition, and have a regular structure, witnessed by the low variance in node degrees, as opposite to the random graphs. This regularity in the graph structure seems to confuse the coloring algorithm by inducing an uniform search space, with no clue for the search