Matching and Independence Complexes Related to Small Grids

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 $\mathrm{Ind}(\Delta_n^m)$ 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 $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.

We write Γ n to denote the 2 by n + 2 grid graph, e.g. Γ 3 is isomorphic to: We define D n := L(Γ n ), e.g. D 3 is isomorphic to In an unpublished manuscript [11], Jonsson establishes basic results regarding the matching complexes for Γ n and more general grid graphs. For example, Jonsson shows that the homotopical depth of M (Γ n ) is ⌈2n/3⌉, which implies that this skeleton of the complex is a wedge of spheres. However, Jonsson states [11, page 3] that "it is probably very hard to determine the homotopy type of" matching complexes of grid graphs.
In [5], Bousquet-Mélou, Linusson, and Nevo introduce the tool of matching trees for the study of independence complexes. In this paper, we will use matching trees to produce a Morse matching on the face poset of M (Γ n ) = Ind(D n ). Our matching algorithm has a recursive structure that allows us to enumerate the number and dimension of cells in a cellular complex homotopy equivalent to Ind(D n ). We use this recursion to determine topological properties of Ind(D n ).
Our techniques actually apply to independence complexes of a larger class of graphs that include D n . Before introducing these graphs, we define two families of related graphs. First, for m ≥ 1 and n ≥ 1, let Y m n denote the extended star graph with a central vertex of degree m and paths of with n edges emanating outward. We refer to one of these paths as a tendril. (We ignore the degenerate cases m = 0 and n = 0.) For example, Y 1 n ∼ = P a n+1 , Y 2 n ∼ = P a 2n+1 , and Y 3 4 is isomorphic to the following: We further define Y m n to be two vertices connected by m disjoint paths each having n + 1 edges. (We ignore the degenerate cases m = 0 and n = 0.) For example, Y 1 n ∼ = P a n+2 , Y 2 n ∼ = C 2n+2 , and Y 3 4 is isomorphic to the following: We will impose a specific labeling on this graph throughout this paper: the leftmost vertex is a, the rightmost vertex is b, and the k-th vertex away from a on the j-th path is (j, k). Let ∆ m n denote the (labeled) graph Y m n+1 with n additional vertices labeled {1, . . . , n} and edges {k, (j, k)} and {k, (j, k + 1)} for each j ∈ [m] and each k ∈ [n]. For example, ∆ 4 3 is isomorphic to In accordance with this numbering scheme, we define ∆ m 0 := Y m 1 and ∆ m −1 := K 1 where K 1 denotes an isolated vertex with no loops. It is straightforward to verify that ∆ 2 n = D n , and hence ∆ m n is a family generalizing D n . The article is structured as follows. In Section 2 we review discrete Morse theory and matching trees for independence complexes. In Section 3 we describe a matching tree procedure for Ind(∆ m n ) which we call the Comb Algorithm. This matching tree produces a cellular chain complex X m n that is homotopy equivalent to the simplicial chain complex for Ind(∆ m n ). In Section 4 we use the Comb Algorithm to establish enumerative properties regarding dimensions of the chain spaces of X m n . Finally, in Section 5 we apply these enumerative results to derive homological properties of Ind(∆ m n ). We conclude with two questions for further research.

Discrete Morse Theory
In this section we introduce tools from discrete Morse theory. Discrete Morse theory was introduced by R. Forman in [10] and has since become a standard tool in topological combinatorics. The main idea of (simplicial) discrete Morse theory is to pair cells in a simplicial complex in a manner that allows them to be cancelled via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex, cellular but possibly non-simplicial, with fewer cells. Further details regarding the following definitions and theorems can be found in [12] and [14].
Definition 2.1. A partial matching in a poset P is a partial matching in the underlying graph of the Hasse diagram of P , i.e. it is a subset µ ⊆ P ×P such that and no c satisfies a < c < b, and • each a ∈ P belongs to at most one element in µ.
When (a, b) ∈ µ, we write a = d(b) and b = u(a). A partial matching on P is called acyclic if there does not exist a cycle with n ≥ 2 and all b i ∈ P being distinct.
Given an acyclic partial matching µ on P , we say that the unmatched elements of P are critical. The following theorem asserts that an acyclic partial matching on the face poset of a polyhedral cell complex is exactly the pairing needed to produce our desired homotopy equivalence. The space ∆ is homotopy equivalent to a cell complex ∆ c with c i cells of dimension i for each i ≥ 0, plus a single 0-dimensional cell in the case where the empty set is paired in the matching.
In [5], Bousquet-Mélou, Linusson, and Nevo introduced matching trees as a way to apply discrete Morse theory to Ind(G) for a simple graph G = (V, E). For a vertex p ∈ V (G), let N (p) denote the neighbors of p in G. A matching tree τ (G) for G is a directed tree constructed according to the following algorithm.  Refer to v as a splitting vertex of τ (G). The node Σ(∅, ∅) is called the root of the matching tree, while any nonroot node of outdegree 1 in τ (G) is called a matching site of τ (G) and any non-root node of outdegree 2 is called a splitting site of τ (G). Note that the empty set is always matched at the last node of the form Σ(∅, B).
A key observation from [5] is that a matching tree on G yields an acyclic partial matching on the face poset of Ind(G) as follows.
Theorem 2.4 ([5], Section 2). A matching tree τ (G) for G yields an acyclic partial matching on the face poset of Ind(G) whose critical cells are given by the non-empty sets Σ(A, B) labeling non-root leaves of τ (G). In particular, for such a set Σ(A, B), the set A yields a critical cell in Ind(G).

The Comb Matching Algorithm
We begin by determining the homotopy type of Ind(Y m n ) and Ind( Y m n ). Since Y m n is a tree for m ≥ 1 and n ≥ 0, we know by work of Ehrenborg and Hetyei [9] that Ind(Y m n ) is either contractible or homotopy equivalent to a single sphere.
Proof. Case 1: n = 3k. We use induction on m. If m = 1, then Y 1 n ∼ = P a 3k+1 ; hence, Ind(Y 1 n ) is contractible [14,Prop 11.16]. Suppose the induction hypothesis holds for ℓ < m. Select a tendril of Y m n and label the vertices 1 through n starting at the leaf. We consider a matching tree on Ind(Y m n ). Perform Step 2 of the MTA with p = 1 and v = 2. Repeat with p = 4 and v = 5 and so on modulo 3. Since n = 3k, we will eventually perform Step 2 with p = n − 2 and v = n − 1. The remaining subgraph of Y m n from which we may select vertices is isomorphic to Y m−1 n . Since Ind(Y m−1 n ) is contractible by assumption, by induction Ind(Y m n ) is contractible as well. Case 2: n = 3k + 1 or n = 3k + 2. Let a be the vertex of degree m in Y m n . We again consider a matching tree on Ind(Y m n ). We apply Step 3 of the MTA with v = a. At the Σ({a}, N (a)) and Σ(∅, {a}) nodes, the remaining subgraphs of Y m n from which we may select vertices are isomorphic to an m-fold disjoint union of P a n−1 's and an m-fold disjoint union of P a n 's respectively. When n = 3k + 1, the union of P a n 's is contractible [14,Prop 11.16], and each subcomplex Ind(P a n−1 ) contributes n−2 3 + 1 = k vertices toward a single critical cell. In total, the vertex a and the vertices from each Ind(P a n−1 ) factor combine to form a single critical cell of dimension mk. When n = 3k + 2, the union of the P a n−1 's is contractible [14,Prop 11.16], and each subcomplex Ind(P a n ) contributes n−1 3 + 1 = k + 1 vertices toward a single critical cell. In total, the vertices from each Ind(P a n ) factor combine to form a single critical cell of dimension m(k + 1) − 1. This gives the result.
Proof. In Y m n , label the two vertices of degree m as a and b respectively. We consider a matching tree on Ind( Y m n ). First, we apply Step 3 of the MTA with v = b. At the Σ({b}, N (b)) and Σ(∅, {b}) nodes, the remaining subgraphs of Y m n from which we may select vertices are isomorphic to Y m n−1 and Y m n respectively. For n = 3k and n = 3k + 1, the result is immediate from applying Lemma 3.1 as one of the branches will produce contractible information.
For the n = 3k + 2 case with m ≥ 3, Lemma 3.1 only shows that two cells of the appropriate dimension exist, but they may not necessarily form a wedge. This is sufficient for the remainder of the article, but we prove that the two cells do, in fact, form a wedge for sake of completeness. Given the matching tree defined above for Ind( Y m n ), let τ denote the cell of dimension mk+1, and let σ denote the cell of dimension m(k+1)−1. In the style of [ , then x i and x i+1 are matched in the matching tree and so b was designated as a free vertex during some application of Step 1 of the MTA. This is not possible as b is included in A∪B in all tree nodes except for the root. If x i > x i+1 , then x i+1 ⊆ x i as sets. This contradicts that b / ∈ x i and b ∈ x i+1 . Consequently, no such generalized alternating path can exist between σ and τ . The feasibility region of σ does not contain τ , and so σ and τ form a wedge per [15,Theorem 2.2].
We now develop a matching tree for Ind(∆ m n ).
The remaining subgraph of ∆ m n from which we may query vertices is isomorphic to Y m k−1 ∆ m n−(k+1) . Since Ind(Y m k−1 ) is known, we can determine the number and dimension of critical cells below this node by inductively applying this algorithm to ∆ m n−(k+1) .
Step 6: At the Σ(∅, {1, 2, . . . , n}) leaf, the remaining subgraph of ∆ m n from which we may query vertices is isomorphic to Y m n+1 . Since Ind( Y m n+1 ) is known, we can determine the number and dimension of critical cells arising below this node. Definition 3.4. Denote by X m n the cellular complex arising from the Comb Algorithm applied to Ind(∆ m n ) for m ≥ 2 and n ≥ 1. As we cannot apply the Comb Algorithm to Ind( We call this process for generating a matching tree for Ind(∆ m n ) the "Comb Algorithm" because of the visual shape of the resulting matching tree. Steps 1 and 2 produce the backbone of the "comb," while Steps 3 through 6 produce the teeth. For example, applying Steps 1 and 2 of the comb algorithm to Ind(∆ m 4 ) leads to the following (partial) matching tree.
n . Let C 0 n denote one less than the number of 0-dimensional cells in X m n . Since the Comb Algorithm will always pair the empty set with a 0-cell, we have C −1 n = 0. In this context, C d n = 0 if d < 0 or n < 0. Recall that the simplicial join of two abstract simplicial complexes ∆ and Γ is the abstract simplicial complex ∆ * Γ = {σ ∪ τ |σ ∈ ∆, τ ∈ Γ}. It is clear from this definition that Ind where a summand is zero if the subscript or superscript is negative. First, whenever k − 1 ≡ 0 mod 3, Ind(Y m k−1 ) is contractible and, consequently, so is Ind(Y m k−1 ) * Ind(∆ m n−(k+1) ). Thus, C d n (k) = 0 when k − 1 ≡ 0 mod 3, and we may assume that k = 3ℓ or k = 3ℓ+2 for some non-negative integer ℓ. Also, note that Ind(Y m k−1 ) * Ind(∆ m n−(k+1) ) is contractible for k = n as Ind(∆ m −1 ) is contractible, i.e. C d n (n) = 0. These observations subsume Steps 3 and 5 of the Comb Algorithm.
Next, we consider C d n (2). Such a d-cell must correspond to the set of d + 1 vertices consisting of the vertex 2, a single vertex contributed from Ind(Y m 1 ), and d − 1 vertices contributed from Ind(∆ m n−3 ). Therefore, the d- . Observe that if d < m + 1, then C d n (3) = 0. Lastly, we simultaneously consider C d n (k) for k ∈ {4, 5, . . . , n, ∅}. As before, we can disregard k ≡ 1 mod 3 and k = n. First, assume that k = 3ℓ for some positive integer ℓ, which implies that Ind(Y m k−1 ) ≃ S mℓ−1 . Now, consider C d n (k). A d-cell contributed from the factor Ind(Y m k−1 ) * Ind(∆ m n−(k+1) ) consists of the vertex k, mℓ vertices from Ind(Y m k−1 ), and d − mℓ vertices from Ind(∆ m n−(k+1) ), provided d−mℓ > 0. We observe that the related factor Ind(Y m Let χ m n denote the reduced Euler characteristic of X m n . Note that since C 0 n is one less than the number of zero-dimensional cells in X m n , we have Proof. Fix m and n as above. Using the recursion (1) for C d n , we obtain The fifth equality in the above list is obtained via reindexing and the observations that C d−(m+1) n−4 = 0 for d < m and C d−m n−3 = 0 for d < m − 1. Corollary 4.4. When m is even, χ m n satisfies the recursion a n = a n−3 − a n−2 − a n−1 with initial conditions a 0 = 1, a 1 = −2, and a 2 = 1, and hence has generating function Proof. Assume that m ≥ 2 is even. First, observe that χ m 0 = 1, χ m 1 = −2, and χ m 2 = 1 by Proposition 4.1, so both relations have the same initial conditions. We can easily verify that χ m 3 = 2 = 1− (−2)− 1 = a 0 − a 1 − a 2 = a 3 . Now, for fixed n, assume that χ m ℓ satisfies both relations for ℓ < n.  When m = 2, the dimensions of C d n have an interesting enumerative interpretation. The sequence A201780 in OEIS [1] is the Riordan array of which can be alternatively defined by with initial conditions T (0, 0) = 1, T (1, 0) = 0, T (2, 0) = 1, and T (j, k) = 0 if k < 0 or j < k.
Proof. The initial conditions of C n d are realized as entries in this Riordan array as follows. We have C 0 0 = 1 = T (2, 0), and Thus, the recursion applied to T (n − d + 2, 3d − 2n) matches that of C n d . The proof of the second half of the claim is similar and omitted. .
Then, C d n = 0 if 0 ≤ d < d min n , excluding the base 0-cell. When m = 2, these two formulas coincide.
Observe that these four values are greater than or equal to 2k + 1 as m ≥ 2. Therefore, none of the cells in X m n are of dimension smaller than 2n+2 3 . Further, when j = 2, we have that the factor Ind(Y m 1 ) * Ind(∆ m n−3 ) produces a cell of dimension exactly 2k + 1.
Observe that these three values are greater than or equal to 2k + m as m ≥ 2. Therefore, none of the cells in X m n are of dimension smaller than 2 n−1 3 + m. Further, when j = 2, we have that the factor Ind(Y m 1 ) * Ind(∆ m n−3 ) produces a cell of dimension exactly 2k + m.
Remark 5.2. Theorem 5.1 shows that X m n is at least d min n -connected. After a suitable adjustment of notation, this agrees with results of Jonsson [11,Proposition 2.7] regarding the connectivity of Ind(∆ 2 n ). .
Proof. By Proposition 4.1, the claim holds for n ∈ {0, 1, 2, 3}. We proceed by induction. Assume n ≥ 4, and suppose the claim is true for all 0 ≤ i < n. Consider the maximum dimension of a cell produced below the node Σ({j}, N (j) ∪ {1, 2, . . . , j − 1}) from the Comb Algorithm applied to Ind(∆ m n ). As before, we may assume j ∈ {2, . . . , n}. If j = n, the remaining subgraph of ∆ m n from which we may query vertices is isomorphic to Y m n+1 . If j < n, then the remaining subgraph is Y m j−1 ∆ m n−(j+1) , which corresponds to a subcomplex of Ind(∆ m n ) of the form Ind(Y m j−1 ) * Ind(∆ m n−(j+1) ). We will again use the notation δ j from the proof of Theorem 5.1.
Proof. Recall from Theorem 5.1 that for n = 3k+2 and m ≥ 3, the minimum dimension of critical cells produced by the Comb Algorithm is 2 n−1 3 + m. It is easy to verify that 2n+2 3 = 2k + 2 < 2k + m = 2 n−1 3 + m. Therefore, C dn n = 0 when n = 3k + 2, i.e. H dn (X m n ; Z) is trivial. Now, assume that n = 3k. We know that C ℓ n = 0 for ℓ < d n from our cellular dimension range. We argue by induction on k that C dn n = 1 while C dn+1 n = 0, which proves the claim for n = 3k. Begin by recalling that C 0 0 = 1 and C 1 0 = 0, which provides a base case. Assume that C d 3ℓ Hence, C dn n = 1.
Assume that n = 3k +1; this argument is similar to the previous case. We again argue by induction on k that C dn n = 1 while C dn+1 n = 0. We obtain our base case by recalling that C 1 1 = 1 and C 2 1 = 0 for m ≥ 4. Next, we know that = 0. Hence, C dn n = 1. We also know that C dn+1 For other homology groups, the Comb Algorithm provides less comprehensive results. For example, when m = 2, that is, when X m n is homotopy equivalent to the matching complex on the 2 × (n + 2) grid graph, a direct analysis of the chain space dimensions on a data table yields the following. As n grows larger, the data suggests that the rank of this particular chain space ceases to "typically" exceed the sum of the ranks of the neighboring chain spaces. This suggests that the behavior of Ind(∆ m n ) for "small" values of n, including many values of n for which by-hand computations appear prohibitive, is not indicative of the general behavior of these complexes.
Thus, the topology of Ind(∆ m n ) remains generally mysterious. It would be of interest to investigate the following two questions.
(1) Does torsion occur in the homology of Ind(∆ m n )? If so, for which p does Z/pZ appear as a summand? (2) There is a natural action of the symmetric group S m on Ind(∆ m n ). What is the S m -module structure of H * (Ind(∆ m n ); C)?