Solution of Problem # 10769

We review previous solutions of problem nr. 10769 posed by Christian Blatter A 1988 paper by C. D. Godsil and J. Zaks [1] states the following result, which is the solution of problem nr. 10769: “Let G be the graph with the points of the unit sphere in three dimensions as its vertices, and two vertices adjacent if and only if they are orthogonal as unit vectors (i.e., their spherical distance is π=2. The chromatic number of G is four.” A proof that four colors suffice for G is constructive and rather elementary. Consider first the intersection points of the sphere with the the x , the y and the z axis, colored by green, blue and red, respectively. There are exactly three great circles which pass through two of these three pairs of points. The great circles can be colored with the two colors used on the four points they pass through. The three great circles divide the sphere into eight open octants of equal area. Four octants, say, in the half space z > 0, are colored by the four colors red, white, green and blue. The remaining octants obtain their color from their antipodal octant. Although the paper is not entirely specific, it is easy to write down an explicit coloring scheme according to the above prescription. Consider spherical coordinates: let θ be the angle between the z axis and the line connecting the origin and the point, and φ be the angle between the x axis and the projection of the line connecting the origin and the point onto the x y plane. In terms of these coordinates, an arbitrary point on the unit sphere is given by (θ;φ;r = 1) (θ;φ).