Emanuele Delucchi
Consider the poset of partitions of {1,...(n-1)k+1} with block sizes congruent to 1 modulo k. We prove that its order complex is a subdivision of the complex of k-trees, thereby answering a question posed by Feichtner. The result is obtained by an ad-hoc generalization of concepts from the theory of nested set complexes to non-lattices.
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The $h$-polynomial of the barycentric subdivision of any $n$-dimensional cubical complex with nonnegative cubical $h$-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type $B_n$. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker.
A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d<4. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in $\mathbb{R}^d$ is simplicially collapsible after d-2 derived subdivisions at most. This presents progress on an old question by Good…
For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair (Q, π), where Q is a word in the alphabet of simple reflections, $π$ is a group element. We discuss the transformations of such a complex induced by braid moves of the word Q. We show that under certain conditions, this transformation is a composition of edge subdivisions and inverse edge subdivisions. In such a case, we describe how the H- and the γ-polynomials change under this operation. This case includes all braid moves for groups with simply-laced Coxeter diagrams.
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…
We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extend Stanley and Reisner's face ring for abstract simplicial complexes and Stanley's face ring for rational fans. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisi…
The subdivision of hypergraphs
本文提出超图细分的概念，证明其可归结为单纯复形细分，并验证了相对于嵌入同调的拓扑不变性。
Notions of sesquiconstructible complexes and odd iterated stellar subdivisions are introduced, and some of their basic properties are verified. The Charney-Davis conjecture is then proven for odd iterated stellar subdivisions of sesquiconstructible balls and spheres.
Unprojection theory aims to analyze and construct complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of non-acyclic Gorenstein simplicial complexes correspond to unprojections of type Kustin-Miller. As an application, we inductively calculate the minimal graded free resolutions of Stanley-Reisner rings associated to stacked polytopes, recovering results of Terai, Hibi, Herzog and Li Marzi.
We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations,…
Assume sigma is a face of a Gorenstein* simplicial complex D. We investigate the question of whether the Weak Lefschetz Property of the Stanley-Reisner ring k[D] (over an infinite field k) is equivalent to the same property of the Stanley-Reisner ring k[D_sigma] of the stellar subdivision D_sigma. We prove that this is the case if the dimension of sigma is big compared to the codimension.
Forbidden subdivision in integral trees
该研究针对整树的结构性质，证明了所有特征值为整数的树不存在含7个顶点的细分边，为整树的结构刻画提供了新的禁用子结构结论。
Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local $h$-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the $h$-vector of a simplicial complex is played by the (short or long) cubical $h$-vector of a cubical complex, defined by Adin, and the role of the local $h$-vector of a triangulation of a simplex is played by the (short or long) cubical local $h$-vector of a cubical subdivision of a cube. The cubical local $h$-vectors are defined in this paper and are shown to …
In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set P_r of r-element multichains from P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there exactly 2^r such functions which yield subdivisions of the order complex of P of which 2^{r-1} are pairwise different. Within this class are for example the order complexes of the interval and the zig-zag poset of P and the rth edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are …
In this paper we show that the pipe dream complex associated to the permutation 1n(n-1)...2 can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck polynomial specializes to the h-polynomial of the corresponding pipe dream complex, which in certain cases equals the h-polynomial of canonical triangulations of root (and flow) polytopes, which in turn equals a specialization of the reduced form of a monomial in the subdivision algebra of root (and flow) polytopes. Thus, we connect Grothendieck polynomials to…
The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for oriented matroid duality.
The enumerative theory of simplicial subdivisions (triangulations) of simplicial complexes was developed by Stanley in order to understand the effect of such subdivisions on the $h$-vector of a simplicial complex. A key role there is played by the concept of a local $h$-vector. This paper surveys some of the highlights of this theory and some recent developments, concerning subdivisions of flag homology spheres and their $γ$-vectors. Several interesting examples and open problems are discussed.
Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric mot…
Alexander's conjecture for infinite simplicial complexes
将Alexander猜想从有限单纯复形推广到无限单纯复形，证明任意两个具有共同子划分的无限三角剖分必有共同星形子划分。
On total domination subdivision numbers of trees
本文研究了树的全控制细分数问题，提出了全控制细分数为3的树的全新刻画方法，扩展了Haynes等人先前的构造性结果。
The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers tha…
A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $γ(G)$ be the minimum cardinality among all dominating sets in $G.$ A graph $G$ is $γ$-$q$-{\it critical} if the smallest subset of edges whose subdivision necessarily increases $γ(G)$ has cardinality $q.$ In this paper we consider mainly $γ$-$q$-critical trees and give some general properties of $gamma$-$q$-critical graphs. In particular, we show that if $T$ is a $γ$-$q$-critical tree, then $1 \leq q \leq n(T)-1$ and we characterize extremal trees when $q=n(T)-1.$ Since a subdivisio…
Reduction trees are a way of encoding a substitution procedure dictated by the relations of an algebra. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We explain how a shelling of the canonical triangulation can be read off from the corresponding reduction tree in the subdivision algebra. We then introduce the notion of shellable reduction trees in the subdivision and related algebras and define h-polynomials of reduction trees. In the case of the subdivision algebra, the h-polynomials of the canonical triangulatio…
Constructive characterizations concerning total outer-independent domination in subdivision trees
本文研究了子图中总外独立支配数的构造性特征，通过分析树的结构确定达到上下界条件的树类。
Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree $\mathcal{T}$ using two types of well-known operations, first-order subdivision and ($1,m$)-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth…
We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is almost everywhere constant. When $K$ is finite, we study the limit face polynomial of $Sd^d(K)$ after F.Brenti-V.Welker and E.Delucchi-A.Pixton-L.Sabalka.
The problem of how to estimate diffusion on a graph effectively is of importance both theoretically and practically. In this paper, we make use of two widely studied indices, geodesic distance and mean first-passage time ($MFPT$) for random walk, to consider such a problem on some treelike models of interest. To this end, we first introduce several types of operations, for instance, $m$th-order subdivision and ($1,m$)-star-fractal operation, to generate the potential candidate models. And then, we develop a class of novel techniques based on mapping for calculating the exact formulas of the bo…
We exhibit an identity of abstract simplicial complexes between the well-studied complex of trees and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of $k$-trees by Hanlon, and we propose yet another, in the context of nested set complexes mo…
For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at $p=1/n$. We consider a natural analogue of this question for higher-dimensional random complexes $X^k(n,p)$, first studied by Cohen, Costa, Farber and Kappeler for $k=2$. Improving previous results, we show that $p=Θ(1/\sqrt{n})$ is the (coarse) threshold for containing a subdivision of any fixed…
Shelling of links and star clusters in edgewise subdivision of a simplex
研究通过边细分单纯形，揭示链接与星簇的组合结构，利用新的排列统计量解释$h$-向量并推导公式。
Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In codimension one, we use canonical filtrations of $\text{Sd}^d (K)$ to upper estimate these limits and get a monotony theorem which makes it possible to improve these estimates given any packing of disjoint simplices in $\text{Sd}^d (K)$. We then introduce a notion of tiling of simplicial complexes having the property that skeletons and barycentric subdivisions of til…
In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice ${\mathcal L}$ we describe a formal deformation of the barycentric subdivision of the atom crosscut complex $Γ({\mathcal L})$ to its order complex $Δ(\bar{\mathca…
A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficientl…
In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G \circ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.
In 2007, Eickmeyer et al. showed that the tree topologies outputted by the Neighbor-Joining (NJ) algorithm and the balanced minimum evolution (BME) method for phylogenetic reconstruction are each determined by a polyhedral subdivision of the space of dissimilarity maps ${\R}^{n \choose 2}$, where $n$ is the number of taxa. In this paper, we will analyze the behavior of the Neighbor-Joining algorithm on five and six taxa and study the geometry and combinatorics of the polyhedral subdivision of the space of dissimilarity maps for six taxa as well as hyperplane representations of each polyhedral …
We introduce and study additive posets. We show that the top homology group (with coefficients in Z/2Z) of a finite dimensional CW-complex carries a structure of an additive poset invariant under subdivisions. Applications to CW-complexes and graphs are discussed.
Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole
本文定义了时钟图，并证明大树宽的无时钟图必然包含完全图、墙的剖分或其线图作为大树宽的基本障碍导出子图。
Repeated Lefschetz-like decompositions for flag doubly Cohen--Macaulay simplicial complexes and gamma vectors of flag spheres
本文提出了旗双重Cohen-Macaulay单纯复形h多项式的重复Lefschetz型分解，通过迭代双重悬垂和边细分净非负集合，建立了旗球面gamma向量与其几何构造的直接联系，并利用布尔与非布尔区别衡量旗球面偏离交叉多胞体边界的程度。
This thesis aims to give the reader an introduction and overview of the cd-index of a poset, as well as establish some new results. We give a combinatorial proof of Ehrenborg and Karu's cd-index subdivision decomposition for Gorenstein* complexes and extend it to a wider class of subdivisions. In doing so, we define a local cd-index that behaves analogously to the well studied local h-vector. We examine known cd-index and h-vector bounds, and then use the local cd-index to bound a particular class of polytopes with the cd-index of a stacked polytope. We conclude by investigating the h-vector a…
In this paper we consider a family of simplicial complexes, which we call the view complexes. Our choice of objects of study is motivated by theoretical distributed computing, since the view complex is a key simplicial construction used for protocol complexes in the snapshot computational model. We show that the view complex $\view$ can be collapsed to the well-known complex $χ(Δ^n)$, called standard chromatic subdivision of a simplex, and that $χ(Δ^n)$ is itself collapsible. Furthermore, we show that the collapses can be performed simultaneously in entire orbits of the natural symmetric group…
It is shown that if T is a connected nontrivial graph and X is an arbitrary finite simplicial complex, then there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials, and subdivision are established that may be of independent interest.
Flag complex face structures and decompositions
本文利用组合方法与切比雪夫展开研究了任意旗单纯球伽马向量的分量结构，揭示了其正性性质，并探讨了重心细分下复形 $f$-向量的分解与几何变换间的相互作用。
In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $Δ_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that $Δ_T$ is $(n-b-1)$-connected when the tree has $b$ leaves. We provide a shelling for the $(n-b)$-skeleton of $Δ_T$, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree $T$ which imply shellability of $Δ_T$, and we construct such inversion functions for a large enough class of trees to ded…
We generalize the concept of combinatorial nested set complexes to posets and exhibit the topological relationship between the arising nested set complexes and the order complex of the underlying poset. In particular, a sufficient condition is given so that this relationship is actually a subdivision. We use the results to generalize the proof method of Čukić and Delucchi, so far restricted to semilattices, for a result of Björner, Paffenholz, Sjöstrand and Ziegler on the Bier construction on posets.
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifte…
We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a …
Lovasz's striking proof of Kneser's conjecture from 1978 using the Borsuk--Ulam theorem provides a lower bound on the chromatic number of a graph. We introduce the shore subdivision of simplicial complexes and use it to show an upper bound to this topological lower bound and to construct a strong Z_2-deformation retraction from the box complex (in the version introduced by Matousek and Ziegler) to the Lovasz complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several ``intermediate'' complexes.
Partitons of vertices and facets in trees and stacked simplicial complexes
本文为堆叠单纯复形（含树、多边形三角剖分、堆叠多面体）建立了面划分与顶点独立集划分之间的显式双射，并将其推广到带最小距离约束的情形，进而导出自然数划分的间距下界结果。
It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical points. The proof is stated in terms of discrete Morse functions on a class of posets that is slightly broader than the class of face posets of finite regular CW complexes.
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.…
Given a finite simplicial complex $X$ and a connected graph $T$ of diameter $1$, in \cite{anton} Dochtermann had conjectured that $\text{Hom}(T,G_{1,X})$ is homotopy equivalent to $X$. Here, $G_{1, X}$ is the reflexive graph obtained by taking the $1$-skeleton of the first barycentric subdivision of $X$ and adding a loop at each vertex. This was proved by Dochtermann and Schultz in \cite{ds12}. In this article, we give an alternate proof of this result by understanding the structure of the cells of Hom$(K_n,G_{1,X})$, where $K_n$ is the complete graph on $n$ vertices. We prove that the neighbo…
Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree
本文研究大树宽有界度图的不可避免导出子图，证明了两个支持Aboulker等人猜想的定理：一是大树宽有界度图必含t-theta、t-pyramid或墙细分线图作为导出子图，二是必含给定次立方细分毛毛虫或其细分线图作为导出子图。
An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a ti…
The dual graph $Γ(h)$ of a regular triangulation $Σ(h)$ carries a natural metric structure. The minimum spanning trees of $Γ(h)$ recently proved to be conclusive for detecting significant data signal in the context of population genetics. In this paper we prove that the parameter space of such minimum spanning trees is organized as a polyhedral fan, called the MST-fan of $Σ(h)$, which subdivides the secondary cone of $Σ(h)$ into parameter cones. We partially describe its local face structure and examine the connection to tropical geometry in virtue of matroids and Bergman fans.
Contractible independence complexes of trees
研究树的独立复形可收缩条件，通过分支点的截断操作将其简化为特定路径，同时探讨独立多项式在-1处的取值特性。
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.
We give new interpretations of the $ν$-Tamari lattice of Préville-Ratelle and Viennot. First, we describe it as a rotation lattice of $ν$-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux and north-east fillings. Then, using a formulation in terms of bracket vectors of $ν$-trees and componentwise order, we provide a simple description of the lattice property. We also show that the $ν$-Tamari lattice is isomorphic to the increasing-flip poset of a suitably chosen subword complex, and settle a special case of Rubey's lattice conjecture concerning the …
Attempting to recognize a tree inside a phylogenetic network is a fundamental undertaking in evolutionary analysis. In the last few years, therefore, tree-based phylogenetic networks, which are defined by a spanning tree called a subdivision tree, have attracted attention of theoretical biologists. However, the application of such networks is still not easy, due to many problems whose time complexities are not clearly understood. In this paper, we provide a general framework for solving those various old or new problems from a coherent perspective, rather than analyzing the complexity of each …
A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.
Comparing the face rings of a boolean complex and its barycentric subdivision
本文研究布尔复形及其重心重分的面环的等变同构问题，证明一般情况不成立，仅在特定Cohen–Macaulay复形下成立并给出显式同构构造。
A matching complex of a simple graph $G$ is a simplicial complex with faces given by the matchings of $G$. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa showed that matching complexes of forests are contractible or homotopy equivalent to a wedge of spheres. We study two specific families of trees. For caterpillar graphs, we give explicit formulas for the number of spheres in each dimension and for perfect binary trees we find a strict connectivity bound. We also use a tool from discrete Morse theory called the \t…
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