Please use this identifier to cite or link to this item:
https://idr.l2.nitk.ac.in/jspui/handle/123456789/12910
Title: | Separation dimension and sparsity |
Authors: | Alon, N. Basavaraju, M. Chandran, L.S. Mathew, R. Rajendraprasad, D. |
Issue Date: | 2018 |
Citation: | Journal of Graph Theory, 2018, Vol.89, 1, pp.14-25 |
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 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(k lg lg n) and that there exists a family of 2-degenerate graphs with separation dimension ?(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 edges. We do not believe that this bound is optimal and give a question and a remark on the optimal bound. 2018 Wiley Periodicals, Inc. |
URI: | http://idr.nitk.ac.in/jspui/handle/123456789/12910 |
Appears in Collections: | 1. Journal Articles |
Files in This Item:
There are no files associated with this item.
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.