Sperner's colorings of hypergraphs arising from edgewise triangulations
Du\v{s}ko Joji\'c, Ognjen Papaz

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 , the hypergraph whose hyperedges are facets of the edgewise triangulation of a -simplex defined by a permutation . Mirzakhani and Vondr\' ak showed that the greedy coloring of produces the maximal number of monochromatic hyperedges. The line graph of is built from the copies of the graph that represents which subsets of consecutive numbers of are contiguous in . We characterize these graphs in terms of dissections a regular -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 on a regular -gon and graphs extends on the group of permutations . Independent sets of the graphs of the…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Graph Theory Research · Limits and Structures in Graph Theory
