Sperner's colorings of hypergraphs arising from edgewise triangulations

We investigate Sperner's labelings of $H^\pi_{k,q}$, the hypergraph whose hyperedges are facets of the edgewise triangulation of a $(k-1)$-simplex defined by a permutation $\pi\in \mathbb{S}_{k-1}$. Mirzakhani and Vondr\' ak showed that the greedy coloring of $H^{\mathrm{Id}}_{k,q}$ produces the maximal number of monochromatic hyperedges. The line graph of $H_{k,q}^\pi$ is built from the copies of the graph $G_\pi$ that represents which subsets of consecutive numbers of $[k-1]$ are contiguous in $\pi$. We characterize these graphs in terms of dissections a regular $k$-gon and also show how they encode the adjacency relation between a hypersimplex and the facets of its alcoved triangulation. The natural action of the dihedral group $D_k$ on a regular $k$-gon and graphs $G_\pi$ extends on the group of permutations $\mathbb S_{k-1}$. Independent sets of the graphs $G_\pi$ of the permutations that are not invariant under the rotation are used to define a class of Sperner's colorings that produce more monochromatic hyperedges then the greedy colorings. This colorings are also optimal for a certain permutations.