New Results on the Queens_n2 Graph Coloring Problem

Given an n×n chess board, a queen graph is a graph with n vertices, each corresponding to a square of the board. Two vertices are connected by an edge if the corresponding squares are in the same row, column, or diagonals (both ascending and descending diagonals), this corresponds to the queen move rule at the chess game. The coloring problem on this graph consists in finding the minimum number of colors necessary for placing n queens on the board so that no two queens of the same color can attack each other. Finding this number (the chromatic number χ) is an optimization problem. We may also consider the following decision problem: given a n chess board, is it possible to place n sets (each corresponding to a given color) of n queens on the board so that there is no clash between two queens in the same set? Gardner [3] states without proof that this is the case if and only if n is not divisible by either 2 or 3. If so, n=χ(Queens n2) –noted χn– since maximum clique number is n. Until now, no chromatic numbers are available for n>9 if n is a multiple of 2 or 3 (see: [1, 2, 4] for recent works).