Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Journal of Algorithms
Constructive generation of very hard 3-colorability instances
Discrete Applied Mathematics
3-coloring in time O (1.3289n)
Journal of Algorithms
The very particular structure of the very hard instances
AAAI'96 Proceedings of the thirteenth national conference on Artificial intelligence - Volume 1
MCPR'11 Proceedings of the Third Mexican conference on Pattern recognition
Computing infeasibility certificates for combinatorial problems through Hilbert's Nullstellensatz
Journal of Symbolic Computation
Using hajós' construction to generate hard graph 3-colorability instances
AISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Symbolic Computation
Recognizing 3-colorings cycle-patterns on graphs
Pattern Recognition Letters
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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.