Given a graph G on the vertex set V, the non-matching complex of G, denoted by ${\mathsf{NM}}_k(G)$, is the family of subgraphs $G^{\prime }\subset G$ whose matching number $\nu (G^{\prime })$ is strictly less than k. As an attempt to extend 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 ${\mathsf{NM}}_k(G)$, which vanishes in all dimensions $d\ge 3k-4$, and all dimensions $d\ge 2k-3$ when G is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes a result by Aharoni, Berger, Chudnovsky, Howard and Seymour: Let $E_1,\dots ,E_{3k-2}$ be non-empty edge subsets of a graph and suppose that $\nu (E_i\cup E_j)\ge k$ for every $i\ne j$. Then $E=\bigcup E_i$ has a rainbow matching of size k. Furthermore, the number of edge sets $E_i$ can be reduced to $2k-1$ when E is the edge set of a bipartite graph.

给定顶点集V上的图G ， G的非匹配复数，表示为${\mathsf{纳米}}_{ķ}(G)$, 是子图族$G^{\text{'}}\subset G$谁的匹配号码$\nu (G^{\text{'}})$严格小于k。作为尝试扩展 Linusson、Shareshian 和 Welker 关于同伦类型的结果${\mathsf{纳米}}_{ķ}({ķ}_n)$和${\mathsf{纳米}}_{ķ}({ķ}_{r,s})$对于任意图G，我们证明 (i)${\mathsf{纳米}}_{ķ}(G)$是$(3ķ-3)$-Leray，并且 (ii) 如果G是二分的，那么${\mathsf{纳米}}_{ķ}(G)$是$(2ķ-2)$-勒雷。这个结果是通过分析复合体的非空面链接的同源性得到的${\mathsf{纳米}}_{ķ}(G)$，在所有维度上消失$d\ge 3ķ-4$, 和所有维度$d\ge 2ķ-3$当G是二分的。作为推论，我们有以下彩虹匹配定理，它概括了 Aharoni、Berger、Chudnovsky、Howard 和 Seymour 的结果：${乙}_1,\dots ,{乙}_{3ķ-2}$是图的非空边子集并假设$\nu ({乙}_{\mathrm{一世}}\cup {乙}_j)\ge ķ$对于每个$\mathrm{一世}\ne j$. 然后$乙=\bigcup {乙}_{\mathrm{一世}}$有大小为k的彩虹匹配。此外，边集的数量${乙}_{\mathrm{一世}}$可以简化为$2ķ-1$当E是二部图的边集时。