Abstract
Lovasz proved that the chromatic number of the graph formed by the principal diagonals of an $$n$$ -dimensional strongly self-dual polytope is greater than or equal to $$n+1$$ . There is equality if the length of the principal diagonals is greater than the Euclidean diameter of the monochromatic parts of that coloring of the unit sphere which is based on a partition of $$n+1$$ congruent spherical regular simplices. We determine this quantity for all $$n$$ and prove that in dimension three all such graphs can be colored by four colors.
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