Cubical Subdivisions and Local h-Vectors

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 share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed.     

Some combinatorial properties of flag simplicial pseudomanifolds and spheres

A simplicial complex Δ is called flag if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension d−1, then the graph of Δ (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 Δ of dimension d−1 is minimized when Δ is the boundary complex of the d-dimensional cross-polytope.     

<Emphasis Type="Italic">f</Emphasis>-Vectors of barycentric subdivisions

For a simplicial complex or more generally Boolean cell complex Δ we study the behavior of the f- and h-vector under barycentric subdivision. We show that if Δ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney–Davis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary complex of a simple polytope. For a general (d − 1)-dimensional simplicial complex Δ the h-polynomial of its n-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this h-polynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d. F. Brenti and V. Welker are partially supported by EU Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272 and the program on “Algebraic Combinatorics” at the Mittag-Leffler Institut in Spring 2005.     

The γ-vector of a barycentric subdivision

We prove that the γ-vector of the barycentric subdivision of a simplicial sphere is the f-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex.     

Signable posets and partitionable simplicial complexes

The notion of apartitionable simplicial complex is extended to that of asignable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termedtotal signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes. The research of S. Onn was supported by the Alexander von Humboldt Stifnung, by the Fund for the Promotion of Research at the Technion, and by Technion VPR fund 192–198.     

A classification of the face numbers of Buchsbaum simplicial posets

The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The \(h'\) -vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the \(h'\) -vector of a Buchsbaum simplicial poset satisfies certain simple inequalities; in this paper we show that these necessary conditions are in fact sufficient to characterize the \(h'\) -vectors of Buchsbaum simplicial posets with prescribed Betti numbers.     

<Emphasis Type="Italic">f</Emphasis>-vectors of simplicial posets that are balls

Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vectors. We also present new methods for constructing poset balls with specific h-vectors. Combining this work with a new result of S. Murai we are able to give a complete characterization of the h-vectors of simplicial poset balls in all even dimensions, as well as odd dimensions less than or equal to five.     

On the Stanley ring of cubical complex

We investigate the properties of the Stanley ring of a cubical complex, a cubical analogue of the Stanley-Reisner ring of a simplicial complex. We compute its Hilbert series in terms of thef-vector, and prove that by taking the initial ideal of the defining relations, with respect to the reverse lexicographic order, we obtain the defining relations of the Stanley-Reisner ring of the triangulation via “pulling the vertices” of the cubical complex. Applying an old idea of Hochster we see that this ring is Cohen-Macaulay when the complex is shellable, and we show that its defining ideal is generated by quadrics when the complex is also a subcomplex of the boundary complex of a convex cubical polytope. We present a cubical analogue of balanced Cohen-Macaulay simplicial complexes: the class of edge-orientable shellable cubical complexes. Using Stanley's results about balanced Cohen-Macaulay simplicial complexes and the degree two homogeneous generating system of the defining ideal, we obtain an infinite set of examples for a conjecture of Eisenbud, Green, and Harris. This conjecture says that theh-vector of a polynomial ring inn variables modulo an ideal which has ann-element homogeneous system of parameters of degree two, is thef-vector of a simplicial complex.     

On the f- and h-triangle of the barycentric subdivision of a simplicial complex

For a simplicial complex Δ we study the behavior of its f- and h-triangle under the action of barycentric subdivision. In particular we describe the f- and h-triangle of its barycentric subdivision sd(Δ). The same has been done for f- and h-vector of sd(Δ) by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the h-triangle of Δ are nonnegative, then the entries of the h-triangle of sd(Δ) are also nonnegative. We conclude with a few properties of the h-triangle of sd(Δ).     

On subdivisions of simplicial complexes: Characterizing localh-vectors

A well-known combinatorial invariant of simplicial complexes is theh-vector, which has been the subject of much combinatorial research. This paper deals withlocal h-vectors, recently defined by Stanley as a tool for studyingh-vectors of simplicial subdivisions. The face-vector of any simplicial complex can only increase when the complex is subdivided; how does theh-vector change? Motivated by this question, Stanley derived certain useful properties of localh-vectors. In this paper we use mainly geometric arguments to show that these properties characterize localh-vectors, andregular localh-vectors.     

A Short Simplicial h -Vector and the Upper Bound Theorem

Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.     

Relations on generalized degree sequences

We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley.     

A Short Simplicial h -Vector and the Upper Bound Theorem

   Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.     

On Generalized h -Vectors of Rational Polytopes with a Symmetry of Prime Order

We prove tight lower bounds for the coefficients of the generalized h -vector of a rational polytope with a symmetry of prime order that is fixed-point free on the boundary. These bounds generalize results of Stanley and Adin for the h -vector of a simplicial rational polytope with a central symmetry or a symmetry of prime order, respectively. Received November 6, 1997, and in revised form March 17, 1998.     

On $$\gamma $$ γ - and local $$\gamma $$ γ -vectors of the interval subdivision

Journal of Algebraic Combinatorics - We show that the $$\gamma $$ -vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In...     

Even and odd holes in cap‐free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β‐perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even‐signable. Graphs that can be signed so that every triangle is odd and every triangle is odd and every hole is odd are called odd‐signable. We derive from a theorem due to Truemper co‐NP characterizations of even‐signable and odd‐signable graphs. A graph is strongly even‐signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even‐signable graph is even‐signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd‐signable. Every strongly odd‐signable graph is odd‐signable. We give co‐NP characterizations for both strongly even‐signable and strongly odd‐signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (cap‐free graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well‐studied subclass of cap‐free graphs. If a graph is strongly even‐signable or strongly odd‐signable, then it is cap‐free. In fact, strongly even‐signable graphs are those cap‐free graphs that are even‐signable. From our decomposition theorem, we derive decomposition results for strongly odd‐signable and strongly even‐signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999     

Squarefree P-modules and the cd-index

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen–Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen–Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley–Reisner ring of the barycentric subdivision of an odd dimensional Cohen–Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function.     

Balanced complexes and complexes without large missing faces

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 piecewise linear (d−1)-sphere is 2(d−l)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.     

The Stirling Polynomial of a Simplicial Complex

We introduce a new encoding of the face numbers of a simplicial complex, its Stirling polynomial, that has a simple expression obtained by multiplying each face number with an appropriate generalized binomial coefficient. We prove that the face numbers of the barycentric subdivision of the free join of two CW-complexes may be found by multiplying the Stirling polynomials of the barycentric subdivisions of the original complexes. We show that the Stirling polynomial of the order complex of any simplicial poset and of certain graded planar posets has non-negative coefficients. By calculating the Stirling polynomial of the order complex of the r-cubical lattice of rank n + 1 in two ways, we provide a combinatorial proof for the following identity of Bernoulli polynomials: