Jared Culbertson, Anton Dochtermann, Dan P. Guralnik, Peter F. Stiller
We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection to linear quotients of quadratic monomial ideals.
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Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure strongly shellable complexes. Meanwhile, pure strongly shellable complexes can be characterized by the corresponding codimension one graphs. In addition, we show that the facet ideals of pure strongly shellable complexes have linear quotients.
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable. Further, we explicitly determine the homology of $q$-complexes corresponding to uniform $q$-matroids. We also outline some partial results concerning the determination of homology of arbitrary shellable $q$-complexes..
Shellability of Higher Independence Complexes of Graphs
本文研究图的高阶独立复形的可壳性分类问题，针对块图、树等图族，基于直径和禁用子图建立可壳性充分条件，并通过星-团粘附、团加须等图操作给出构造性方法，推广了经典独立复形的相关结果。
The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the crosspolytopes allow such a shelling. Furthermore, we show that the complex of directed trees of a complete double directed graph is a union of suitable spheres. We also investigate shellability of the maximal pure skeleton of a complex of directed trees.…
Shellability of relative squeezed balls and spheres
研究相对挤压球及其边界复形的可剥壳性问题，通过证明其可剥壳，给出明确剥壳顺序并刻画限制面，为这类新复形建立强大组合结构。
We investigate the shellability of the polyhedral join $\mathcal{Z}^*_M (K, L)$ of simplicial complexes $K, M$ and a subcomplex $L \subset K$. We give sufficient conditions and necessary conditions on $(K, L)$ for $\mathcal{Z}^*_M (K, L)$ being shellable. In particular, we show that for some pairs $(K, L)$, $\mathcal{Z}^*_M (K, L)$ becomes shellable regardless of whether $M$ is shellable or not. Polyhedral joins can be applied to graph theory as the independence complex of a certain generalized version of lexicographic products of graphs which we define in this paper. The graph obtained from t…
We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the $CL$-shellability criterion of Björner and Wachs for posets and its generalization by Kozlov called $CC$-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank $2n$ by the action of the wreath product $S_2\wr S_n$ of symmetric groups, and we provide a partitioning for the quotient complex $Δ(Π_n)/S_n $. Stanley asked for a description of the symmetric group representation $β_S $ on the homology of the rank-selected partitio…
Shellability of a simplicial complex has many useful structural implications. In particular, it was shown by Danaraj and Klee that every shellable pseudo-manifold is a PL-sphere. The purpose of this paper is to prove the shellability of the quasi-arc complex of the Möbius strip. Along the way we provide elementary proofs of the shellability of the arc complex of the $n$-gon and the cylinder. In turn, applying the result of Danaraj and Klee, we obtain the sphericity of all of these complexes.
For a property $\cal P$ of simplicial complexes, a simplicial complex $Γ$ is an obstruction to $\cal P$ if $Γ$ itself does not satisfy $\cal P$ but all of its proper restrictions satisfy $\cal P$. In this paper, we determine all obstructions to shellability of dimension $\le 2$, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions $\le 2$. We also show that these three sets of obstructions coincide in the class of flag complexes. These results …
Shellability is a well-known combinatorial criterion for verifying that a simplicial complex is Cohen-Macaulay. Another notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, was shown to imply Golodness of the associated Stanley-Reisner ring. The two algebraic notions were tied together by Herzog, Reiner and Welker who showed that if the Alexander dual of a complex is …
Shellability is hard even for balls
本文证明当维度至少为3时，三角剖分d-球的可壳化是NP困难的，并扩展证明了3-复形的可坍缩性和2-复形的可壳化在特定条件下的NP困难性。
In this paper, we give a new algebraic criterion for the {\em shellability} of (non-pure) simplicial complex $Δ$ over $[n]$, shellable in the sense of Björner and Wachs \cite{BW}. We show that the spanning simplicial complex of doubly uni-cyclic graph is non-pure shellable. Moreover, we introduce the concept of Gallai-simplicial complex $Δ_Γ(G)$ of a finite simple graph $G$. We applied the obtained criterion to discuss the shellability of Gallai simplicial complexes associated to various classes of graphs..
Let $Φ$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $Δ^m (Φ)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $Δ^m (Φ)$ is shellable and by V. Reiner that it is $(m+1)$-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part $Δ^m_+ (Φ)$ of $Δ^m (Φ)$. An explicit homotopy equivalence is given between $Δ^m_+ (Φ)$ and the poset of generalized noncrossing partitions, associated to the pair $(Φ, m)$ by D. Armstro…
Shellability in Clique-Free Complexes of Graphs
本文研究了$t$-无团复形的壳性，证明了特定条件下的复形可分解且壳性，并引入了团附加操作以构造壳性复形，同时扩展了团尾巴的概念并建立了弦性与线性解的关系。
Chordality, syzygies, and shellability for hypergraphic analogues of interval graphs
本文证明共区间超图与下闭复形在补集意义下等价，并证明下闭团簇是Woodroofe弦性的，从而回答Dochtermann和Engström的问题，表明其Alexander对偶复形是顶点可分解的，且具有由下闭顶点序诱导的壳化。
Shellability of $3$-Cut Complexes of Squared Cycle Graphs
本文证明了平方圈图的 $3$-割复形是壳的，并验证了其 Betti 数的猜想。
Shellability of 3-cut complexes of powers of cycle graphs
研究3-切割复形Δ_3(C_n^p)的壳性，证明其同调等价于多个球面的 wedge，提供显式壳序并计数生成元数量。
A lattice framework for generalizing shellable complexes and matroids
该研究引入幂格概念统一推广多种格结构，定义P-复形推广可剥性概念，构造推广型拟阵并证明相关可剥性结论，还给出多重集格拟阵构造方法及一类Stanley-Reisner环的序列Cohen-Macaulay性质。
Homotopy type of shellable $q$-complexes and their homology groups
本文针对可剥$q$-单纯复形，确定了其同伦型，证明了含$q$-拟阵复形的字典序可剥$q$-单纯复形的序复形可剥，进而全面确定了这类复形的同调群，拓展了$q$-复形可剥性的拓扑研究。
On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs
本文证明了六边形网格图的3-割复形对于所有m,n≥1都是壳性的，并给出了壳序构造及球面数量的计数公式。
Complexes of pattern-avoiding injective words
本文通过固定置换模式定义单射词复形的自然子复形，证明其在多数模式下的可壳性并构造同调基，同时利用枚举组合学证明可壳性，并推导出模式回避置换的新的精细计数公式。
We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional, simplicial complex is $k$-decomposable is NP-hard. For $d \ge 3$, both problems remain NP-hard when restricted to contractible pure $d$-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.
We prove that the second derived subdivision of any rectilinear triangulation of any convex polytope is shellable. Also, we prove that the first derived subdivision of every rectilinear triangulation of any convex 3-dimensional polytope is shellable. This complements Mary Ellen Rudin's classical example of a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is a new relative notion of shellability that characterizes the behavior of shellable complexes under gluing. As a corollary, we obtain a new characterization of the PL property in terms of shellability: A triangulat…
The shellability status of previously investigated simplicial complexes with up to 24 facets is settled. In case of shellability the exact number of shellings is determined. Our algorithm merely relies on the facets, and not on additional information such as the face-numbers.
Topics in Boolean Representable Simplicial Complexes
本文研究了布尔可表示单纯复形（BRSC）理论，探讨了其上的各种算子，并考察了更高维度的可壳性，同时提出了若干新猜想。
In extremal graph theory, the problem of finding the elements of a given class of graphs which contain the most cliques traces its routes back to Turán's famous theorem. We consider the implications of the connectivity property of simplicial complexes known as shellability on clique complexes associated with graphs. In this paper, we find the graphs which maximize cliques among all graphs on $n$ vertices with $Δ(G)\leq r$ that have shellable clique complexes.
Weighted simplicial complexes (WSCs) are powerful tools for describing weighted cloud data or networks with weighted nodes. In this paper, we propose a novel approach to study WSCs via the concept of polarization. Polarization of a WSC allows one to construct a new (unweighted) simplicial complex which coincides with an object called the mixed wreath product. This new construction preserves several properties and invariants of the underlying simplicial complex of a WSC. Our main focus is to analyze WSCs through their underlying simplicial complexes and mixed wreath products. Combinatorially, w…
We prove a new theorem of Tverberg type which confirms the conjecture of Blagojevic, Frick, and Ziegler about the existence of "balanced Tverberg partitions" (Conjecture 6.6 in, Tverberg plus constraints, Bull. London Math. Soc., 46 (2014) 953-967). The proof relies on the connectivity and shellability properties of multiple chessboard complexes and their symmetric analogues.
Given an arbitrary hypergraph $\mathcal{H}$, we may glue to $\mathcal{H}$ a family of hypergraphs to get a new hypergraph $\mathcal{H}'$ having $\mathcal{H}$ as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph $\mathcal{H}'$ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying ni…
We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon's identity is re-established by constructing an infinite family of non-pure simplicial complexes $Δ(n)$, indexed by the positive integers, such that the alternating sum of the numbers of faces of $Δ(n)$ of each dimension is the left-hand side of the identity. We show that $Δ(n)$ is shellable for all $n$. Then, using the fact that a shellable simplicial complex is homotopy equivalent to a wedge of spheres, we compute the Betti numbers of $Δ(n)$ by counting (via a generating function…
Linearly shellable complexes
本文定义线性可剥壳纯单纯复形，证明多类 Coxeter 复形满足该性质，并提出线性强可剥壳性概念。
Extendability of $1$-decomposable complexes
本文研究了1-可分解复形的扩展性，证明了任何纯的d维1-可分解复形可以扩展到特定的骨架，同时保持1-可分解性。
Shellings of Unbounded Polyhedra
研究无界多面体边界复形的剥壳性，先研究紧化，利用多面体结果得热带超曲面剥壳性，探讨多面体复形对偶性与剥壳性相互作用，转化到离散莫尔斯理论，揭示正则细分紧支撑特性。
The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier. Björner, Paffenholz, Sjöstrand and Ziegler generalize this construction to obtain a Bier poset Bier(P,I) from any bounded poset P and any proper ideal I of P. They show shellability of Bier(P,I) for the case where P is the boolean lattice, and obtain thereby 'many shellable spheres' in the sense of Kalai. We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov. We obtain 'more shellable spheres' by proving the general statement that combinatorial …
Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. Shellings of a complex expose a large amount of information in a helpful way, but are not easy to construct, often requiring deep information about the structure of the complex. It is natural to ask whether shellings may be efficiently found computationally. In a recent paper, Goaoc, Paták, Patáková, Tancer and Wagner gave a negative answer to this question (assuming P \neq NP), showing that the problem of deciding whether a simplicial complex is shellable is NP-complete. In this paper, w…
We recently defined a property of Morse shellability (and tileability) of finite simplicial complexes which extends the classical one and its relations with discrete Morse theory. We now prove that the product of two Morse tileable or shellable simplicial complexes carries Morse tileable or shellable triangulations under some tameness condition, and that any tiling or shelling becomes tame after one barycentric subdivision. We deduce that any finite product of closed manifolds of dimensions less than four carries Morse shellable triangulations whose critical and h-vectors are palindromic. We a…
If a pure simplicial complex is partitionable, then its $h$-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex $Δ$, we construct a complex $Γ\supseteq Δ$ of the same dimension such that both $Γ$ and the relative complex $(Γ,Δ)$ are partitionable. This allows us to rewrite the $h$-vector of any pure simplicial complex as the difference of two $h$-vectors of partitionable complexes, giving an analogous interpretation of the $h$-vector of a non-partitionable complex. By contrast, for a given complex $Δ$ it is not always possibl…
It is proved that fundamental groups of boolean representable simplicial complexes are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\geq 2$. In the case of dimension 2, it is shown that boolean representable simplicial complexes have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen-Macaulay are determined. Complexity bounds are provided for all the algorithms involved.
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable comp…
Vertex Dismissibility and Scalability of Simplicial Complexe
本文研究了单纯复形的顶点可弃性和可扩展性，揭示了其严格层次结构及代数结构，通过拓扑与同调视角统一了经典结果。
Topology of Cut Complexes of Graphs
本文定义了图的k-切割复形，推广了Alexander对偶，并利用代数拓扑、离散Morse理论等技术研究了多种图族的可壳性、同伦型和同调性质。
Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths
本文研究根系偏序集反链构成的单纯复形的可壳性，证明其等价于根系为A_n、B_n、D_3或G_2，并利用f-和h-三角形给出Dyck路径的统计量，有理情形留作开放问题。
On the Vertex Decomposability of $r$-Independence Complexes of Trees
本文证明了对于任意树 $T$ 和整数 $r \ge 1$，$r$-独立复形 $\Ind_r(T)$ 是顶点可分解的，从而解决了 Abdelmalek 等人的一个猜想。
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