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Publication Type : Journal Article
Publisher : Graphs and Combinatorics
Source : Graphs and Combinatorics
Url : https://link.springer.com/article/10.1007/s00373-020-02163-y
Campus : Coimbatore
School : School of Engineering
Year : 2020
Abstract : In this paper, we study the embedding of a complete balanced d-partite d-uniform hypergraph with its nd vertices represented as points in general position in Rd and each hyperedge drawn as the convex hull of d corresponding vertices. We assume that the set of vertices is partitioned into d disjoint sets, each of size n, such that each vertex in a hyperedge is from a different set. Two hyperedges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. Using Colored Tverberg theorem with restricted dimensions, we observe that such an embedding of a complete balanced d-partite d-uniform hypergraph with nd vertices contains Ω((8/3)d/2)(n/2)d((n−1)/2)d crossing pairs of hyperedges for n≥3 and sufficiently large d. Using Gale transform and Ham-Sandwich theorem, we improve this lower bound to Ω(2d)(n/2)d((n−1)/2)d for n≥3 and sufficiently large d.
Cite this Research Publication : R. Gangopadhyay and S. Shannigrahi. Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs. Graphs and Combinatorics 36, 905-911, 2020.