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The permutahedron $\Pi_n$ is the polytope that is the convex hull of all permutations of the vector $(1,2,...,n)$. There are many results on its structure, but I couldn't find a result on the chromatic number of its 1-skeleton. Is it known?

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    $\begingroup$ It is bipartite; color a permutation with its sign. $\endgroup$
    – David E Speyer
    Commented Jun 2, 2016 at 10:35

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