A vertex coloring of a simplicial complex $Δ$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $Δ$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that $v_1\in F_1\backslash F_2$ and $v_2\in F_2\backslash F_1$. We show that every simplicial complex $Δ$ which is linearly colored with $k$ colors includes a subcomplex $Δ'$ with $k$ vertices such that $Δ'$ is a strong deformation retract of $Δ$. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.
We show that coloring properties of a simplicial complex K are reflected by splitting properties of a bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley-Reisner algebra of K.
We define an (r,s)-coloring of an abstract simplicial complex to be a coloring using r colors of the vertices so that in any simplex at most s vertices have the same color. We translate the problem of finding an (r,s)-coloring of a given simplex to a topological problem and to a ring theoretic problem. The topological version involves vector bundles over Davis--Januszkiewicz spaces and the algebraic one involves factorizations in Stanley-Reisner rings.
For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric algebraic varieties.
We introduce the notion of nonevasive reduction, and show that for any monotone poset map $ϕ:P\to P$, the simplicial complex $Δ(P)$ {\tt NE}-reduces to $Δ(Q)$, for any $Q\supseteq{\text{\rm Fix}}ϕ$. As a corollary, we prove that for any order-preserving map $ϕ:P\to P$ satisfying $ϕ(x)\geq x$, for any $x\in P$, the simplicial complex $Δ(P)$ collapses to $Δ(ϕ(P))$. We also obtain a generalization of Crapo's closure theorem.
In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.
We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric variety. In general, this gives a class of ``anti-Ramsey theory problems'' with a geometric structure. For example, we find one for Ramsey numbers of classes of such graphs. The key observation is that our previous result expressing simplicial chromatic polynomials in terms of $h$-vectors of auxiliary simplicial complexes can be reinterpreted as one on edge c…
We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $Δ$ whenever the number of colors is at least $q\geq (\frac{10}{3} + ε)Δ$, where $ε>0$ is arbitrary and the maximum degree satisfies $Δ\geq C$ for a constant $C = C(ε)$ depending only on $ε$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-ε_0 ) Δ$, where $ε_0 \approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenh…
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like ``holonomy'', ``parallel transport'', ``bundles'', ``combinatorial curvature'' etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial ``Theorema Egregium'' for cubical complexes non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool for extending Babson-Kozlov-Lovasz graph coloring results to more gen…
A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; we propose the balanced analog of the Generalized Lower Bound Conjecture and establish some related results. We close with constructions of balanced manifol…
We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre's condition $(S_{\ell}),$ we show that the equivariant flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector satisfy several inequalities. We apply these results to the study of $P$-partitions of double posets, and weak colorings of mixed graphs.
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley's $(-1)$-color theorem. A $q$-analog version of this family of characters is also studied.
Let $G$ be a simple graph on $d$ vertices. We define a monomial ideal $K$ in the Stanley-Reisner ring $A$ of the order complex of the Boolean algebra on $d$ atoms. The monomials in $K$ are in one-to-one correspondence with the proper colorings of $G$. In particular, the Hilbert polynomial of $K$ equals the chromatic polynomial of $G$. The ideal $K$ is generated by square-free monomials, so $A/K$ is the Stanley-Reisner ring of a simplicial complex $C$. The $h$-vector of $C$ is a certain transformation of the tail $T(n)= n^d-k(n)$ of the chromatic polynomial $k$ of $G$. The combinatorial structu…
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial $d$-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every $m \geq d+2$ and any closed combinatorial $d$-manifold $M$, two $m$-colored triangulations of $M$ can be connected by …
In general a contractible complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one "gets stuck", that is one can choose the collapses in such a way that one arrives at a nontrivial complex which admits no collapsing moves. Here we examine this phenomenon in the case of a simplex. In particular we characterize all values of $n$ and $d$ so that the n-simplex may collapse to a d-complex from which no further collapses are possible. Equivalently and in the language of high-dimensional generalizations of trees, we const…
Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex Delta_{A,H} as the subdivision of the link of A induced by H. In particular, this generalizes Steingrimsson's coloring complex of a graph. We do the following: (1) When A is a hyperplane arrangement, Delta_{A,H} is shown to be shellable. As a special case, we answer affirmatively a question of Steingrimsson on coloring complexes. (2) For H being a Coxeter arrangement of type A or B we obtain a close connection between the Hilbert series of the Stanley-Reisner ring of Delta_{A,…
Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomology of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured…
In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \sqcup T_{2}$, such that each $T_{i}$ represents the lines of a copy of the Fano plane $PG(2, \mathbb{F}_{2})$. We generalize this observation by constructing, for each prime power $q$, a simplicial complex $X_{q}$ with $q^{2} + q + 1$ vertices and $2(q^{2} + q + 1)$ facets consisting of two copies of $PG(2, \mathbb{F}_{q})$. Our construction works for any colored $k$-configuration, defined as a $k$-configuration whose associated bipartite graph $G$ is connected and has a $k$-edge coloring $χ\colon E(G) \…
A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d-1 that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is d-collapsible is NP-complete for d greater or equal to 4 and polynomial time solvable for d at most 2. As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for d at most 2, but the greedy algorithm does not work for d greater or equal 3. A simplicial complex is d-representable if it is …
On colorings of hypergraphs embeddable in $\mathbb{R}^d$
Keimyung University（启明大学）；Einstein Institute of Mathematics, Hebrew University（希伯来大学爱因斯坦数学研究所）；Mathematics Research Institute, Universidad de Valladolid（瓦拉多利德大学数学研究所）；CMSA, Harvard University（哈佛大学数学与统计应用中心）
A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in turn a generalization of classical Catalan combinatorics). The latter paper gave two possible models for a rational analog of the associahedron which attach simplicial complexes to any pair of coprime positive integers a < b. These complexes coincide up to the Fuss-Catalan level of generality, but in general one may be a strict subcomplex of the other. Verif…
