Matching complexes of small grids

The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. It is known that the matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$. In this paper, we study the homotopy type of the matching complex of $(n \times 2)$-grid graph $\Gamma_n$. Braun and Hough introduced a family of graphs $\Delta^m_n$, which is a generalization of $L(\Gamma_n)$. In this paper, we show that the independence complex of $\Delta^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.


Introduction
A matching on a simple graph G = (V (G), E(G)) is a subgraph of G whose matximal degree is at most 1. A matching is identified with its edge set. The matching complex M (G) of G is the simplicial complex whose simplices are the matchings on G. In general, it is very difficult to determine the homotopy types of matching complexes (see [3], [8], and [10]). We refer to [7] for a concrete introduction to this subject.
In particular, we write Γ n instead of Γ(n, 2).
Jonsson first studied the matching complex of grid graphs in his unpublished paper [6]. Recently, Braun and Hough [4] investigated the matching complex of Γ n , and they use discrete Morse theory to derive some homological properties of M (Γ n ). In fact, they studied more general simplicial complexes. To state it precisely, we need some preparation.
For a graph G, the independence complex I(G) of G is the simplicial complex whose simplices are the independent sets of G. The line graph L(G) of G is the graph whose vertex set is E(G), and two distinct edges e and e ′ of G are adjacent if and only if they have a common endpoint. Then the matching complex M (G) coincides with the independence complex of the line graph L(G).   Figure 1 For a pair m and n of positive integers, Braun and Hough [4] introduced the graph ∆ m n , which is a generalization of L(Γ n ). The vertex set of ∆ m n consists of e i for i = 1, · · · , n and f k i for i = 1, · · · , n − 1 and k = 1, · · · , m. The adjacent relations are given as follows: , (i = 1, · · · , n − 1) Figure 2 depicts the graph ∆ 4 5 . Clearly, ∆ 2 n and L(Γ n ) are isomorphic, and hence I(∆ 2 n ) and M (Γ n ) are isomorphic.
Braun and Hough [4] actually studied the independence complexes of ∆ m n . The purpose of this paper is to determine the homotopy types of the independence complex of ∆ m n . The following two theorems are the main results in the present paper.
Here Σ denotes the reduced suspension. In particular, we have  Figure 2.
By Theorem 1.2, the homotopy type of I(∆ m n ) is recursively determined by I(∆ m 1 ), · · · , I(∆ m 4 ). In Section 4, we show that all these complexes are wedges of spheres, and hence we have the following theorem: n is a wedge of spheres for positive integers m and n. In particular, the matching complex M (Γ n ) of Γ n is homotopy equivalent to a wedge of spheres.
In particular, the homology groups of the independence complex of ∆ m n has no torsions. This gives an answer to a problem suggested by Braun and Hough (see the end of [4]).
This paper is organized as follows. In Section 2, we review some facts concerning independence complexes. Since Theorem 1.1 is easily deduced from known results, we discuss it in this section. Theorem 1.2 and Theorem 1.4 are proved in Section 3 and Section 4, respectively.

Preliminaries
We refer to [7] and [9] for fundamental terms and facts concerning simplicial complexes.
We first recall the following simple observation of independence complexes (see [1] for example). For a vertex v of G, the link of v in I(G) coincides with Here I(G) \ v denotes the subcomplex of I(G) whose simplices are the simplices of I(G) not containing v. This observation clearly yields the following proposition: is null-homotopic, then we have Proof. By the above observation, it suffices to see that  Here we give the proof of Theorem 1.1 since it easily follows from Proposition 2.2.
See Figure 3. In this section, we prove Theorem 1.2. Throughout this section, we assume that m is an integer greater than 1. Suppose n ≥ 2, and put X n = ∆ m n \ e n−1 . Since N ∆ m n (e n ) ⊂ N ∆ m n (e n−1 ), Proposition 2.2 implies the following: Lemma 3.1. For n ≥ 2 and m ≥ 2, we have I(∆ m n ) ≃ I(X n ).
Next we consider the graph Y n = X n \ e n−2 (see Figure 5).  Figure 5 is a homotopy equivalence. Note that every vertex of . This means that the composite is null-homotopic. Here we use the assumption m ≥ 2 to show that the first map is a homotopy equivalence. Thus the inclusion I(X n \ N Xn [e n−2 ]) → I(Y n ) is null-homotopic, and this completes the proof.
Next we study the homotopy type of I(Y n ) Proof. We want to apply Proposition 2.1 to the vertex e n of Y n . Namely, we must show the following: (1) The inclusion I(Y n \ N Yn [e n ]) ֒→ I(Y n \ e n ) is null-homotopic.
(2) The homotopy type of I(Y n \ N Yn [e n ]) is Σ m I(∆ m n−4 ). (3) The homotopy type of I(Y n \ e n ) is Σ m I(∆ m n−3 ). Define the induced subgraphs Z n , Z ′ n , and Z ′′ n of Y n as follows:  the commutative diagram that the inclusion I(Z n ) ֒→ I(Y n ) is null-homotopic. By the sequence of inclusions, we have that I(Y n \ N Yn [e n−3 ]) ֒→ I(Y n ) is null-homotopic. This completes the proof of (1). Finally, we prove (3). By Proposition 2.2, it is easy to see that I(Y n \ e n ) is homotopy equivalent to I(W n ) (see Figure 6). Here W n is defined by . This completes the proof of (3 In this section, we prove Theorem 1.4, which asserts that I(∆ m n ) is a wedge of spheres. The case m = 1 is proved by Theorem 1.1, and hence we assume m ≥ 2 in the rest of this section. It follows from Theorem 1.  For m ≥ 2, the complexes I(∆ m 1 ), · · · , I(∆ m 4 ) are described as follows: Proof. Note that I(∆ m 1 ) is a point. It clearly follows from Proposition 2.2 that I(∆ m 2 ) ≃ I(K 2 ) = S 0 .
Consider the case of n = 3. By Lemma 3.1, we have that I(∆ m 3 ) ≃ I(X 3 ). Braun and Hough determined the homotopy types of the independence complexes of X 3 (see Lemma 3.2 of [4]), but we give an alternative proof of this result for self-containedness. First Proposition 2.2 implies that I(X 3 \ e 3 ) and I(X 3 \ {e 1 , e 3 }) are homotopy equivalent. Since X 3 \ {e 1 , e 3 } is the m-copies of K 2 , we have I(X 3 \ e 3 ) ≃ I(X 3 \ {e 1 , e 3 }) = S m−1 .
On the other hand, applying Proposition 2.2 again, we have that I(X 3 \ N X 3 [e 3 ]) and I(K 2 ) = S 0 are homotopy equivalent. Since every map from S 0 to S m−1 is null-homotopic, the inclusion I(X 3 \ N X 3 [e 3 ]) ֒→ I(X 3 \ e 3 ) is null-homotopic. Thus Proposition 2.1 implies I(X 3 ) = S 1 ∨ S m−1 .
Finally we consider the case n = 4. By Proposition 3.2 and I(∆ m 1 ) = * , we have that I(X 4 ) ≃ I(Y 4 ). By Proposition 2.2, I(Y 4 \ e 4 ) is homotopy equivalent to the independence complex of the disjoint union of one isolated vertex and m-copies of K 2 , and hence contractible. In particular, the inclusion I(