Sperner's colorings of hypergraphs arising from edgewise triangulations
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arXiv
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| Natura: | Preprint |
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2025
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| author | Jojić, Duško Papaz, Ognjen |
| author_facet | Jojić, Duško Papaz, Ognjen |
| contents | We investigate Sperner's labelings of $H^π_{k,q}$, the hypergraph whose hyperedges are facets of the edgewise triangulation of a $(k-1)$-simplex defined by a permutation $π\in \mathbb{S}_{k-1}$. 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 $H_{k,q}^π$ is built from the copies of the graph $G_π$ that represents which subsets of consecutive numbers of $[k-1]$ are contiguous in $π$. 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 $G_π$ extends on the group of permutations $\mathbb S_{k-1}$. Independent sets of the graphs $G_π$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07201 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sperner's colorings of hypergraphs arising from edgewise triangulations Jojić, Duško Papaz, Ognjen Combinatorics 05C15, 05E18, 05C30, 05A05 We investigate Sperner's labelings of $H^π_{k,q}$, the hypergraph whose hyperedges are facets of the edgewise triangulation of a $(k-1)$-simplex defined by a permutation $π\in \mathbb{S}_{k-1}$. 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 $H_{k,q}^π$ is built from the copies of the graph $G_π$ that represents which subsets of consecutive numbers of $[k-1]$ are contiguous in $π$. 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 $G_π$ extends on the group of permutations $\mathbb S_{k-1}$. Independent sets of the graphs $G_π$ 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. |
| title | Sperner's colorings of hypergraphs arising from edgewise triangulations |
| topic | Combinatorics 05C15, 05E18, 05C30, 05A05 |
| url | https://arxiv.org/abs/2506.07201 |