Sperner's colorings of hypergraphs arising from edgewise triangulations

TL;DR

This paper explores Sperner's labelings of hypergraphs derived from edgewise triangulations of simplices, characterizing associated graphs and introducing colorings that outperform greedy methods in producing monochromatic hyperedges.

Contribution

It provides a detailed characterization of the graphs related to Sperner's colorings and introduces new coloring strategies that yield more monochromatic hyperedges than traditional greedy approaches.

Findings

Characterization of graphs $G_$ in terms of polygon dissections

Identification of permutation classes with optimal colorings

Development of Sperner's colorings surpassing greedy methods

Abstract

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.