For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex BNM_k(n) of all bipartite graphs G on bipartition [r] \cup [\bar{s}] such that there is no matching of size k in G, and obtain results similar to those obtained for NM_k(n).
The matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jelić Milutinović et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.
The bounded degree complex $BD(G,λ)$ is a generalization of the matching complexes of a graph. In this paper, we show that the bounded degree complex of a forest is shellable, by using independence complexes of hypergraphs. We obtain a wedge decomposition result of bounded degree complexes when a graph $G$ has a leaf
The matching complex $M(G)$ of a graph $G$ is the set of all matchings in $G$. A Buchsbaum simplicial complex is a generalization of both a homology manifold and a Cohen--Macaulay complex. We give a complete characterization of the graphs $G$ for which $M(G)$ is a two-dimensional Buchsbaum complex. As an intermediate step, we determine which graphs have matching complexes that are themselves connected graphs.
The {\em perfect matching complex} of a graph is the simplicial complex on the edge set of the graph with facets corresponding to perfect matchings of the graph. This paper studies the perfect matching complexes, $\mathcal{M}_p(H_{k \times m\times n})$, of honeycomb graphs. For $k = 1$, $\mathcal{M}_p(H_{1\times m\times n})$ is contractible unless $n\ge m=2$, in which case it is homotopy equivalent to the $(n-1)$-sphere. Also, $\mathcal{M}_p(H_{2\times 2\times 2})$ is homotopy equivalent to the wedge of two 3-spheres. The proofs use discrete Morse theory.
Keevash and Mycroft [\emph{Mem.~Amer.~Math.~Soc., 2015}] developed a geometric theory for hypergraph matchings and characterized the dense simplicial complexes that contain a perfect matching. Their proof uses the hypergraph regularity method and the hypergraph blow-up lemma recently developed by Keevash. In this note we give a new proof of their results, which avoids these complex tools. In particular, our proof uses the lattice-based absorbing method developed by the author and a recent probabilistic argument of Kohayakawa, Person and the author.
The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper we completely characterize the pairs (graph, matching complex) for which the matching complex is a homology manifold, with or without boundary. Except in dimension two, all of these manifolds are sphere or balls.
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.
A (general) polygonal line tiling is a graph formed by a string of cycles, each intersecting the previous at an edge, no three intersecting. In 2022, Matsushita proved the matching complex of a certain type of polygonal line tiling with even cycles is homotopy equivalent to a wedge of spheres. In this paper, we extend Matsushita's work to include a larger family of graphs and carry out a closer analysis of lines of triangle and pentagons, where the Fibonacci numbers arise.
We investigate the representation of a symmetric group $S_n$ on the homology of its Quillen complex at a prime $p$. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of $p$-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of $S_n$ on the top homology group of the corresponding hypergraph matching complex when $n \equiv 1 \bmod p$. Our conjecture follows from work of Bouc when $p=2$, and we prove the conjecture when $p=3$.
Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the union of collapsible complexes. Also, we prove that all links in these complexes are suspensions up to homotopy. Furthermore, we extend the definition of a cubical matching complex to planar graphs that are not necessarily bipartite, and show that these complexes are either contractible or a disjoint union of contractible complexes. For a simple connected region…
The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(Δ_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(Δ_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(Δ_n^m)$ and prove that several homology groups of $\mathrm{Ind}(Δ_n^m)$ are non-zero.
A $2$-matching complex is a simplicial complex which captures the relationship between $2$-matchings of a graph. In this paper, we will use discrete Morse Theory and the Matching Tree Algorithm to prove homotopical results. We will consider a class of graphs for which the homotopy type of the $2$-matching complex transforms from a sphere to a point with the addition of leaves. We end the paper by defining $k$-matching sequences and looking at the $1$- and $2$-matching complexes of wheel graphs and perfect caterpillar graphs.
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…
Let $L_n$ be a line graph with $n$ edges and $\F(L_n)$ the facet ideal of its matching complex. In this paper, we provide the irreducible decomposition of $\F(L_n)$ and some exact formulas for the projective dimension and the regularity of $\F(L_n)$.
To any finite simplicial complex X, we associate a natural filtration starting from Chari and Joswig's discrete Morse complex and abutting to the matching complex of X. This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph.
The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$. Braun and Hough introduced a family of graphs $Δ^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $Δ^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.
Given a graph $G$ on the vertex set $V$, the non-matching complex of $G$, $\mathsf{NM}_k(G)$, is the family of subgraphs $G' \subset G$ whose matching number $ν(G')$ is strictly less than $k$. As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of $\mathsf{NM}_k(K_n)$ and $\mathsf{NM}_k(K_{r,s})$ to arbitrary graphs $G$, we show that (i) $\mathsf{NM}_k(G)$ is $(3k-3)$-Leray, and (ii) if $G$ is bipartite, then $\mathsf{NM}_k(G)$ is $(2k-2)$-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex $\mathsf…
Addressing a question of Zaremsky, we give conditions on a finite simplicial graph which guarantee that the associated matching arc complex is connected and hyperbolic.
Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vrécica and {Ž}ivaljević established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large $n$, the bottom nonvanishing homo…
For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at most $r$. In this article, we give a closed form formula for the homotopy type of the $(n-2)$-matching complex of complete graph on $n$ vertices. We also prove that the $(n-1)$-matching complex of complete bipartite graph $K_{n,n}$ is homotopy equivalent to a sphere of dimension $(n-1)^2-1$.
Given an arbitrary sequence of non-negative integers $\vecλ=(λ_1,\dots,λ_n)$ and a graph $G$ with vertex set $\{v_1,\dots,v_n\}$, the bounded degree complex, denoted $\text{BD}^{\vecλ}(G)$, is a simplicial complex whose faces are the subsets $H\subseteq E(G)$ such that for each $i \in \{1,\dots,n\}$, the degree of vertex $v_i$ in the induced subgraph $G[H]$ is at most $λ_i$. When $λ_i=k$ for all $i$, the bounded degree complex $\text{BD}^{\vecλ}(G)$ is called the $k$-matching complex, denoted $M_k(G)$. In this article, we determine the homotopy type of bounded degree complexes of forests. In p…
With every matching in a graph we associate a group called the matching group. We study this group using the theory of non-positively curved cubed complexes. Our approach is formulated in terms of so-called gliding systems.
In this article, we discuss the vertex decomposability of three well-studied simplicial complexes associated to forests. In particular, we show that the bounded degree complex of a forest and the complex of directed trees of a multidiforest are vertex decomposable. We then prove that the non-cover complex of a forest is either contractible or homotopy equivalent to a sphere. Finally, we provide a complete characterization of forests whose non-cover complexes are vertex decomposable.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expanders. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders, according to these definitions, exist for d >= 2. We …
It is known that there are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating to the language of $h$-vectors, there are finitely many simplicial complexes of bounded dimension with $h_1=k$ for any natural number $k$. In this paper we study the question at the other end of the $h$-vector: Are there only finitely many $(d-1)$-dimensional simplicial complexes with $h_d=k$ for any given $k$? The answer is no if we consider general complexes, but when focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes…
Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new combinatorial stratification of the Salvetti complex. For the obtained acyclic matchin…
We prove that any quasirandom uniform hypergraph $H$ can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the sparse setting when the density of $H$ quickly tends to $0$ in terms of the number of vertices of $H$. Our results answer and address questions of Kim, Kühn, Osthus and Tyomkyn; and Glock, Kühn and Osthus as well as Keevash. The provided approximate decompositions exhibit strong quasirandom properties which is very useful for forthcoming applications. Our resu…
For a graph $G$, let $ν_s(G)$ be the induced matching number of $G$. We prove that $ν_s(G) \geq \frac{n(G)}{(\lceil\fracΔ{2}\rceil+1) (\lfloor\fracΔ{2}\rfloor+1)}$ for every graph of sufficiently large maximum degree $Δ$ and without isolated vertices. This bound is sharp. Moreover, there is polynomial-time algorithm which computes induced matchings of size as stated above.
A matching $M$ in a graph $G$ is acyclic if the subgraph of $G$ induced by the set of vertices that are incident to an edge in $M$ is a forest. We prove that every graph with $n$ vertices, maximum degree at most $Δ$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{Δ^2},$ and we explain how to find such an acyclic matching in polynomial time.
Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula f…
Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every $d$ there are unbounded degree simplicial complexes of dimension $d$ with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders exist for $d \geq 2$. We present an explicit construction of bounded degree complexes of dimension $d=2$ which are topological…
We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Cohen-Macauley. Finally, we discuss a related series of complexes to provide some positive evidence that the braided Houghton groups, introduced by F. Degenhardt, are a series of groups with linearly increasing finiteness length as are the (unbraided) Houghton groups.
We introduce a new model of random $d$-dimensional simplicial complexes, for $d\geq 2$, whose $(d-1)$-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs [Ke14], and the proof of the expansion uses techniques developed by Evra and Kaufman in [EK15]. This gives a full solution to a question raised in [DK12], which was solved in the two-dimensional case by Lubotzky and Meshulam [LM13].
In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.
We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of the complex onto a ball around this convex subcomplex. These projections will induce a convenient gradient matching on the complex. Besides we analyze the combinatorial structure of both CAT(0) and systolic locally finite simplicial complexes of arbitrary dimensions. We will show that both such complexes possess an arborescent structure. Along the way we make …
We prove a version of Clifford's theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree $2r$ and rank $r$ (for $0<r<g-1$) also carries a divisor of degree $2$ and rank $1$. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens' theorem for metric graphs.
