Matching complexes of $\bf 3 \times n$ grid graphs

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.


Introduction
A matching in a (simple) graph G is a collection of pairwise disjoint edges of G. The matching complex of G, denoted M(G), is a simplicial complex whose vertex set is the edge set of G and simplices are all the matchings in G. The matching complexes first appeared in the 1979 work of Garst [4], where the matching complexes of complete bipartite graphs (also known as the chessboard complexes) were studied while dealing with the Tits coset complexes. In 1992, Bouc [1] studied the matching complexes of complete graphs in connection with the Brown complexes and Quillen complexes. Thereafter, these complexes arose in connection with several areas of mathematics. For a broader perspective, see the 2003 survey article of Wachs [10]. Let [n] denotes the set {1, 2, . . . , n}. For two positive integers m, n, the m × n (rectangular) grid graph Γ m,n is a graph with vertex set V (Γ m,n ) and edge set E(Γ m,n ) defined as follows: V (Γ m,n ) = {(i, j) ∈ N 2 : i ∈ [m], j ∈ [n]}, and E(Γ m,n ) = {((i, j), (i ′ , j ′ )) : |i − i ′ | + |j − j ′ | = 1}.
The matching complex of Γ 1,n was computed by Kozlov in [8]. In the same article, he also computed the matching complexes of cycle graphs. In 2005, Jonsson [6] studied the homotopical depth and topological connectivity of matching complexes of grid graphs and stated that "it is probably very hard to determine the homotopy type of" these complexes. In 2017, Braun and Hough [2] obtained homological results related to matching complexes of 2 × n grid graphs. Matsushita [9], in 2019, extended their results by showing that the matching complexes of 2 × n grid graphs are homotopy equivalent to a wedge of spheres. In this article, we compute the homotopy type of matching complexes of 3×n grid graphs. The main results of this article are summarised below. Theorem 1.1. For n ≥ 1, the matching complex of Γ 3,n is homotopy equivalent to a wedge of spheres. Moreover, if n ∈ {9k, 9k + 1, . . . , 9k + 8} for some k ≥ 0, then where b i 's are some positive integers and ≃ denotes the homotopy equivalence of spaces.
For a graph G, a subset I ⊆ V (G) is said to be independent if there are no edges in the induced subgraph G[I], i.e., E(G[I]) = ∅. The independence complex of G, denoted Ind(G), is a simplicial complex whose vertex set is V (G) and simplices are all the independent subsets of G. The line graph of a graph G, denoted L(G), is a graph with V (L(G)) = E(G) and two distinct vertices (a 1 , b 1 ), (a 2 , b 2 ) ∈ V (L(G)) are adjacent if and only if {a 1 , b 1 } ∩ {a 2 , b 2 } = ∅. Note that the matching complex of G is same as the independence complex of its line graph, i.e., M(G) = Ind(L(G)).
Let G n denotes the line graph of the grid graph Γ 3,n . To compute the homotopy type of M(Γ 3,n ), we determine the homotopy type of Ind(G n ). The main idea used in this article for the computation of Ind(G n ) is to make a step by step careful choice to reduce the graph G n and arrive at different classes of graphs (a total of nine). All these new nine classes of graphs have been defined in Section 3.1. To obtain the Theorem 1.1, we use simultaneous inductive arguments on the independence complexes of these ten classes of graphs. For a quick overview of the relations between all these ten classes of graphs, we refer the reader to see Figure 5.1.
Flow of the article: In the following section, we list out various definitions and results that are used in this article. Section 3 is subdivided into three major subsections. The first two subsections deal with the base cases for the graph G n along with nine more associated classes of graphs. In the next subsection, Section 3.3, we provide and prove recursive formulae to compute the homotopy type of the independence complexes of these ten classes of graphs. The main result of Section 4 is Theorem 4.1, which gives the exact dimensions of the spheres occurring in the homotopy type of the independence complexes of the above mentioned ten classes of graphs.

Preliminaries
An (abstract) simplicial complex K is a collection of finite sets such that if τ ∈ K and σ ⊂ τ , then σ ∈ K. The elements of K are called the simplices (or faces) of K. If σ ∈ K and |σ| = k + 1, then σ is said to be k-dimensional. The set of 0-dimensional simplices of K is denoted by V (K), and its elements are called vertices of K. A subcomplex of a simplicial complex K is a simplicial complex whose simplices are contained in K. In this article, we always assume empty set as a simplex of any simplicial complex and we consider any simplicial complex as a topological space, namely its geometric realization. For the definition of geometric realization, we refer to Kozlov's book [7].
For a simplex σ ∈ K, define The simplicial complexes lk(σ, K) and del(σ, K) are called link of σ in K and (face) deletion of σ in K respectively. The join of two simplicial complexes K 1 and K 2 , denoted as K 1 * K 2 , is a simplicial complex whose simplices are disjoint union of simplices of K 1 and K 2 . Let ∆ S denotes a (|S| − 1)-dimensional simplex with vertex set S. The cone on K with apex a, denoted as C a (K), is defined as For a, b / ∈ V (K), the suspension of K, denoted as Σ(K), is defined as Observe that for any vertex v ∈ V (K), we have Clearly, C v (lk(v, K)) is contractible. Therefore, from [5, Example 0.14], we have the following.
The following observation directly follows from the definition of independence complexes of graphs.
Lemma 2.2. Let G 1 ⊔ G 2 denotes the disjoint union of two graphs G 1 and G 2 . Then . A graph homomorphism is called an isomorphism if it is bijective and its inverse map is also a graph homomorphism. Two graphs G and H are said to be isomorphic if there is an isomorphism between them and we denote it by . For a subset B ⊆ E(G), we let G − B to be the graph with the vertex set V (G − B) = V (G) and the edge set E(G − B) = E(G) \ B. Lemma 2.3. Let G be a graph and {a, b} be a 1-simplex in Ind(G).
For r ≥ 1, the path graph P r is a graph with V (P r ) = [r] and E(P r ) = {(i, i + For r ≥ 3, the cycle graph C r is the graph with V (C r ) = [r] and E(C r ) = {(i, i + 1) : We now proceed towards the main graph of this article. Recall that for m, n ∈ N, the m × n grid graph is denoted by Γ m,n and G n = L(Γ 3,n ) denotes the line graph of Γ 3,n (see Figure 2.1 for example). Formally we define G n with, In this section, we define nine new graph classes viz.
and compute the homotopy type of their independence complexes along with that of G n . The n-th member of each of these graph classes contains a copy of G n as an induced subgraph but not of G n+1 .
We divide this section into three subsections. In the first subsection, we define the above said nine classes of graphs and compute the homotopy type of independence complexes of these graphs along with G n for n = 1. In the next subsection, we compute the homotopy type of independence complexes of all these graphs for n = 2. The final subsection is devoted towards proving recursive formulae for the independence complexes of all ten graph classes, thereby computing their homotopy types. In particular, we show that the independence complex of each of the graphs from these graph classes is a wedge of spheres up to homotopy.
3.1.2. B n . For n ≥ 1, we define the graph B n as follows: For n ≥ 1, we define the graph A n as follows: For n ≥ 1, we define the graph D n as follows: 3.1.5. J n . For n ≥ 1, we define the graph J n as follows: For n ≥ 1, we define the graph J n as follows: 3.1.7. M n . For n ≥ 1, we define the graph M n as follows: 3.1.8. Q n . For n ≥ 1, we define the graph Q n as follows: 3.1.9. F n . For n ≥ 1, we define the graph F n as follows: 3.1.10. H n . For n ≥ 1, we define the graph H n as follows: We consider the vertex b 4 in B 2 (see figure on the right) and analyse del(b 4 , Ind(B 2 )) and lk(b 4 , Ind(B 2 )). First note that lk(b 4 , Ind(B 2 )) = Ind( Figure 3.10c). Since Ind(B 2 )). Hence the inclusion map Ind(B ′′ 2 − {b 2 }) ֒→ del(b 4 , Ind(B 2 )) is null homotopic. Therefore the following composition of maps is null homotopic Ind(B 2 )). Hence lk(b 4 , Ind(B 2 )) is contractible in del(b 4 , Ind(B 2 )) and therefore by Lemma 2.1, Ind(B 2 ))).
3.2.9. F 2 . Since f 3 is a simplicial vertex in F 2 , from Lemma 2.5 we have  Figure 3.13b). Thus using Lemma 2.5, we get 3.3. General case computation. The main outcome of this subsection is that the independence complex of any graph among the ten classes of graphs (defined in Section 3.1) is a wedge of spheres up to homotopy. We prove this by induction on the subscript of the graphs, i.e., n. The cases n = 1, 2 follow from the Section 3.1 and Section 3.2. Fix n ≥ 3, and inductively assume that for any k < n, the independence complex of any graph, among the ten classes of graphs, with subscript k is a wedge of spheres up to homotopy.
Proof. The result follows from Section 3.1.1, Section 3.2.1, induction hypothesis, and Equation (3). 3. 3.2. B n . Let n ≥ 3 and consider the vertex b 4 u 2 ), (u 1 , v 2 )}. Now using the similar arguments as in the case of B 2 , we get that Ind(B n ) ≃ Ind(B ′′ n ). Moreover, the similar arguments imply that Corollary 3.2. For n ≥ 1, Ind(B n ) is homotopy equivalent to a wedge of spheres.

Figure 3.19
We now compute the homotopy type of del(m 1 ,  Proof. The result follows from Section 3.1.7, Section 3.2.7, induction hypothesis, and Equation (9). 3.3.8. Q n . For n ≥ 3, using the same arguments along the lines as in the case of Q 2 we get that, Ind(Q n ) ≃ Σ(Ind(M n )) ∨ Σ 2 (Ind(M n−1 )).
Corollary 3.8. For n ≥ 1, Ind(Q n ) is homotopy equivalent to a wedge of spheres.
(12) Corollary 3.10. For n ≥ 1, Ind(H n ) is homotopy equivalent to a wedge of spheres.

Dimension of the spheres occurring in the homotopy type
In this section we determine the dimensions of all the spheres occurring in the homotopy type of the independence complexes of all ten classes of graphs defined in Section 3.  From Theorem 4.1, we see that the number 9 plays an important role in determining the dimensions of spheres that occur in the homotopy type of M(Γ 3,n ). It would be interesting to see if there is any relation between the number or the dimension of spheres in the homotopy type of M(Γ 3,n ) and the combinatorial description of Γ 3,n . Another interesting enumerative problem is to calculate the Betti numbers of M(Γ 3,n ). More precisely, Question 5.1. Can we determine the closed form formula for the homotopy type of M(Γ 3,n )?
Based on the main result of this article, our computer-based computations for various general grid graphs, and [9], we propose the following. Conjecture 1. The complex M(Γ m,n ) is homotopy equivalent to a wedge of spheres for any grid graph Γ m,n .