Eran Nevo
We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.
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The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal $f$-vector; and moreover, among all such 2-Cohen--Macaulay (2-CM) complexes, the same sphere has the componentwise minimal $h$-vector. It is also verified that the $l$-skeleton of a flag $(d-1)$-dimensional 2-CM complex is $2(d-l)$-CM while the $l$-skeleton of a flag PL $(d-1)$-sphere is $2(d-l)$-homotopy CM. In addition, tight lower bounds …
Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $σ\notin X$ such that $τ\in X$ for any $τ\subsetneq σ$. For a $k$-dimensional simplex $σ$ in $X$, its degree in $X$ is the number of $(k+1)$-dimensional simplices in $X$ containing it. Let $δ_k$ denote the minimal degree of a $k$-dimensional simplex in $X$. Let $L_k$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $μ_k(X)$ denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps $μ_k(X)$, for complexes $X$ without missing faces of dimension larger than $d$: \[…
Lower bounds on the $g$-numbers of spheres without large missing faces
研究无大缺失面球面的g数下界，基于图的独立数推导新界，证明旗球面满足g₂≥(1/2-δ(d))f₀，并展示g向量初始段构成水平序列，进一步得到g数不等式。
Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $μ_{j}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $μ_{k}(X)$ for $k\geq d$ and $μ_{d-1}(X)$. In particular, we establish the following vanishing result: If $μ_{d-1}(X)>(1-\binom{k+1}{d}^{-1})n$, then $\tilde{H}^{j}(X;\mathbb{R})=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval and Montejano f…
Missing faces of neighborly and nearly neighborly polytopes and spheres
本文研究邻接与近邻接多面体和球面的缺失面数，给出近邻接球面的下界，几乎刻画2-邻接4-球面的m-数，并构造缺失面数为零的无穷族及非多面体判定条件。
Simplicial spheres with $g_k=1$
本文研究了具有g_k=1的单纯spheres的结构，针对不同k和d的条件，给出了无缺失面的刻画。
Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be written as the intersection of $k$ $d$-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of $X$ is a set $τ\subset V$ such that $τ\notin X$ but $σ\in X$ for any $σ\subsetneq τ$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension la…
In studying properties of simple drawings of the complete graph in the sphere, two natural questions arose for us: can an edge have multiple segments on the boundary of the same face? and is each face the intersection of sides of 3-cycles? The second is asserted to be obvious in two previously published articles, but when asked, authors of both papers were unable to provide a proof. We present a proof. The first is quite easily proved and the technique yields a third, even simpler, fact: no three edges at a vertex all have internal points incident with the same face.
The only open case of Vizing's conjecture that every planar graph with $Δ\geq 6$ is a class 1 graph is $Δ= 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of $G$ is incident to at most three 3-faces. A stronger statement without restriction to critical graphs is stated in \cite{Wang_Xu_2013}. However, the proof given there works only for critical graphs. Furthermore, we show that every 5-critical plane graph has a 3-face which is adjacent to a $k$-face $(k\in \{3,4\})$. For $Δ= 5$ our result gives insights into the structure …
A Log-Free Lower Bound for the Number of Facets of $0/1$-Polytopes
该研究针对0/1多面体，移除了前人下界中的对数因子，通过随机符号多面体与拉德马赫速率体的比较等方法，得到了更优的面数下界。
We improve Larman's bound on the diameter of a polytope by showing that if $Δ$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $Δ$ is not larger than $2^{r-2}n$, where $n$ is the number of vertices of $Δ$. We then use this result to provide new upper bounds on the diameters of the facet-ridge graphs of $k$-neighborly spheres, $r$-stacked spheres, and polytopes with small $g_r$. Specifically, our bounds imply that $r$-stacked spheres with $r=O(\log n)$ satisfy the polynomial Hirsch conjecture.
Affine stresses: the partition of unity and Kalai's reconstruction conjectures
本文提出了Kalai重构猜想的高维推广，证明了仿射$i$-应力空间在特定条件下决定仿射$1$-应力空间，并建立了仿射应力的单位分解定理。
We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes (they all have at least 3^d nonempty faces) and show that the Hanner polytopes among them (which have exactly 3^d nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3^d+16 nonempty faces.
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