Separation dimension and sparsity

Alon, Noga and Basavaraju, Manu and Chandran, Sunil L and Mathew, Rogers and Rajendraprasad, Deepak (2018) Separation dimension and sparsity. In: JOURNAL OF GRAPH THEORY, 89 (1). pp. 14-25.

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Abstract

The separation dimension(G) of a hypergraph G is the smallest natural number k for which the vertices of G can be embedded in Rk 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(klglgn) and that there exists a family of 2-degenerate graphs with separation dimension (lglgn). 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(4lgn)s-2<bold>n edges</bold>. We do not believe that this bound is optimal and give a question and a remark on the optimal bound.

Item Type:	Journal Article
Publication:	JOURNAL OF GRAPH THEORY
Publisher:	WILEY, 111 RIVER ST, HOBOKEN 07030-5774, NJ USA
Additional Information:	Copyright of this article belong to WILEY, 111 RIVER ST, HOBOKEN 07030-5774, NJ USA
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	23 Jul 2018 15:43
Last Modified:	28 Feb 2019 08:37
URI:	http://eprints.iisc.ac.in/id/eprint/60267

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