Some combinatorial properties of flag simplicial pseudomanifolds and spheres

Abstract

A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $\Delta$ of dimension $d-1$ is minimized when $\Delta$ is the boundary complex of the $d$-dimensional cross-polytope.

Final version, 11 pages. This version, which is somewhat shorter, contains a new result (Theorem 1.2)

References in corpus (2)

• Real Root Conjecture fails for five and higher dimensional spheres
• On the graph-connectivity of skeleta of convex polytopes

Cited by in corpus (8)

• Roundness of grains in cellular microstructures
• On the regularity of edge ideal of graphs
• Face enumeration on flag complexes and flag spheres
• Balanced complexes and complexes without large missing faces
• Connectivity through bounds for the Castelnuovo-Mumford regularity
• Remarks on missing faces and generalized lower bounds on face numbers
• Higher-dimensional counterexamples to Hamiltonicity
• $g$-vectors of manifolds with boundary