Separation index of graphs and stacked 2-spheres

Burton, Benjamin A and Datta, Basudeb and Singh, Nitin and Spreer, Jonathan (2015) Separation index of graphs and stacked 2-spheres. In: JOURNAL OF COMBINATORIAL THEORY SERIES A, 136 . pp. 184-197.

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Abstract

In 1987, Kalai proved that stacked spheres of dimension d >= 3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d = 2. In this article, we give a characterisation of stacked 2-spheres using what we call the separation index. Namely, we show that the separation index of a triangulated 2-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all n-vertex triangulated 2-spheres, the separation index is minimised by some n-vertex flag sphere for n >= 6. Furthermore, we apply this characterisation of stacked 2-spheres to settle the outstanding 3-dimensional case of the Lutz-Sulanke-Swartz conjecture that ``tight-neighbourly triangulated manifolds are tight''. For dimension d >= 4, the conjecture has already been proved by Effenberger following a result of Novik and Swartz. (C) 2015 Elsevier Inc. All rights reserved.

Item Type:	Journal Article
Publication:	JOURNAL OF COMBINATORIAL THEORY SERIES A
Publisher:	ACADEMIC PRESS INC ELSEVIER SCIENCE
Additional Information:	Copy right for this article belongs to the ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
Keywords:	Stacked 2-spheres; Triangulation of 3-manifolds; Tight triangulation; Tight-neighbourly triangulation
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	01 Oct 2015 04:23
Last Modified:	01 Oct 2015 04:23
URI:	http://eprints.iisc.ac.in/id/eprint/52467

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