## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, May 22, 2008, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Eran Nevo (Cornell Univ., Ithaca)

## A characterization of simplicial polytopes with g2=1

Let $v$ and $e$ be the numbers of vertices and edges, respectively, of a simplicial polytope $P$. Define $g_2(P)=e-dv+\binom{d+1}{2}$. Barnette proved that $g_2(P)$ is nonnegative. Kalai proved that the simplicial polytopes with $\g_2=0$ are the stacked polytopes ($d\geq 4$). We characterize the $\g_2=1$ case. It turns out they are the polytopes obtained by stacking over either a join of two simplices, each of dimension at least two, whose dimensions add up to $d$, or a join of a polygon with a $(d-2)$-simplex ($d\geq 4$). The proof uses rigidity theory of graphs and Alexander duality, and works in the generality of homology spheres. Related problems will be mentioned. All the needed notions will be explained in the talk.

Joint work with Eyal Novinsky.

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