Complete and Incomplete Algorithms for the Queen Graph Coloring Problem

The queen graph coloring problem consists in covering a n × n chessboard with n queens, so that two queens of the same color cannot attack each other. When the size, n, of the chessboard is a multiple of 2 or 3, it is hard to color the queen graph with only n colors. We have developed an exact algorithm which is able to solve exhaustively this problem for dimensions up to n = 12 and find one solution for n = 14 in one week of computing time. The 454 solutions of Queens 12 show horizontal and vertical symmetries in the color repartition on the chessboard. From this observation, we design a new exact, but incomplete, algorithm which leads us to color Queens n problems with n colors for n = 15, 16, 18, 20, 21, 22, 24, 28 and 32 in less than 24 hours of computing time by the exploitation of symmetries and other geometric properties.