Separation dimension and sparsity. Issue 1 (1st February 2018)

• Record Type: Journal Article
• Authors: Alon, Noga; Basavaraju, Manu; Chandran, L. Sunil; Mathew, Rogers; Rajendraprasad, Deepak
• Is Part Of: Journal of graph theory. Volume 89:Issue 1(2018)
• Page Start: 14
• Page End: 25
• Publication Date: 2018-02-01
• Subjects: degeneracy -- edge density -- separation dimension; Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.22236
• Languages: English
• ISSNs: 0364-9024

Abstract

Abstract: The separation dimension π ( G ) of a hypergraph G is the smallest natural number k for which the vertices of G can be embedded in R k so that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the cardinality of a smallest family F of total orders of V ( G ), such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. Separation dimension is a monotone parameter; adding more edges cannot reduce the separation dimension of a hypergraph. In this article, we discuss the influence of separation dimension and edge‐density of a graph on one another. On one hand, we show that the maximum separation dimension of a k ‐degenerate graph on n vertices is O ( k lg lg n ) and that there exists a family of 2‐degenerate graphs with separation dimension Ω ( lg lg n ) . On the other hand, we show that graphs with bounded separation dimension cannot be very dense. Quantitatively, we prove that n ‐vertex graphs with separation dimension s have at most 3 ( 4 lg n ) s − 2 · n edges. We do not believe that this bound is optimal and give a question and a remark on the optimal bound.