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.

中文翻译：

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单位球体中的强自对偶多面体和距离图

Lovasz 证明了由 $$n$$ 维强自对偶多胞体的主对角线形成的图的色数大于或等于 $$n+1$$ 。如果主对角线的长度大于基于 $$n+1$$ 全等球面规则单纯形划分的单位球体着色的单色部分的欧几里德直径，则存在相等。我们为所有 $$n$$ 确定这个数量，并证明在维度三中所有这样的图都可以用四种颜色着色。