Separation index of graphs and stacked 2-spheres

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. MSC 2010 : 57Q15, 57M20, 05C40.