We propose a combinatorial approach to the following strengthening of Gal's conjecture: $γ(Δ)\ge γ(E)$ coefficientwise, where $Δ$ is a flag homology sphere and $E\subseteq Δ$ an induced homology sphere of codimension $1$. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag $d$-polytopes: $h_1(Δ) h_i(Δ) \ge (d-i+1)h_{i-1}(Δ) + (i+1) h_{i+1}(Δ)$ for all $0\le i\le d$.
In this note we construct a flag simplicial $3$-sphere $Δ$ with the following properties: - $Δ$ is not a suspension; - $Δ$ has no edge that can be contracted to obtain another flag sphere; - The only equators (induced subcomplexes which are spheres of codimension $1$) of $Δ$ are vertex links. Our construction has $12$ vertices, the minimum number of vertices such a simplicial complex can have. This answers a question posed by Chudnovsky and Nevo.
We give a survey on the recent results and problems on the face enumeration of flag complexes and flag simplicial spheres, with an emphasis on the characterization of face vectors of flag complexes, several lower-bound type of conjectures including the Charney-Davis conjecture and Gal's conjecture, and the upper bound conjecture for flag spheres and pseudomanifolds.
We provide lower and upper bounds on the minimum size of a maximum stable set over graphs of flag spheres, as a function of the dimension of the sphere and the number of vertices. Further, we use stable sets to obtain an improved Lower Bound Theorem for the face numbers of flag spheres.
Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge subdivision or its inverse. For flag spheres we pose new conjectures on their combinatorial structure forced by their face numbers, analogous to the extremal examples in the upper and lower bound theorems for simplicial spheres. Furthermore, we show that our algorithm to test the conjectures searches through the entire space of flag PL spheres of any given …
We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters $(f_0,f_1,f_2,f_3;f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex 4-polytope. The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was foun…
We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Δ$, we show that its $γ$-vector $γ^Δ=(1,γ_1,γ_2,\ldots)$ satisfies: \begin{align*} γ_j=0,\text{ for all } j>γ_1, \quad γ_2\leq\binom{γ_1}{2}, \quad γ_{γ_1}\in\{0,1\}, \quad \text{ and }γ_{γ_1-1}\in\{0,1,2,γ_1\}, \end{align*} supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $Δ$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $γ$-vectors of flag ho…
Planar ternary graphs, flag spheres, and Delannoy polynomials
University of Kansas（堪萨斯大学）；University of Vermont（佛蒙特大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进研究所数学系）；Youngstown State University（扬斯敦州立大学）；Georgia Institute of Technology（佐治亚理工学院）
Sharp Bounds on the Independence Number of Simplicial Spheres
Duke University（杜克大学）；University of California, Davis（加州大学戴维斯分校）；Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）
