论文标题
具有有界匹配号码的图形的复合物数量
Leray numbers of complexes of graphs with bounded matching number
论文作者
论文摘要
给定顶点套装$ v $的图$ g $，$ g $，$ \ mathsf {nm} _k（g）$的非匹配络合物是子级$ g'\子集g $的家族，其匹配的数字$ nmumb $ nmumbs $ nmumn $ nmund $ nmubs $ nmubs $ nmubs $ nment $ k $严格少于$ k $。为了试图概括linusson，shareshian和welker的$ \ mathsf {nm} _k（k_n）$和$ \ m athsf {nm} _k（k__ {k_ {r，s}）$ to nutagry Graphs $ g $，我们显示（我们）$ \ mathssf \ g $ g $ \ mathsf \ $（3K-3）$ - leray，（ii）如果$ g $是双方，则$ \ m athsf {nm} _k（g）$是$（2K-2）$ - leray。通过分析复杂的$ \ mathsf {nm} _K（g）$的非空面条的链接的同源性来获得该结果，该链接在所有方面都消失了$ d \ geq 3k-4 $，以及所有尺寸$ d \ d \ d \ d \ degq 2k-3 $当$ g $ g $ g $ bipArtite时。作为推论，我们具有以下彩虹匹配的定理，该定理概括了Aharoni，Berger，Chudnovsky，Chudnovsky，Howard和Seymour：Let $ e_1，\ dots，e_ {3k-2} $是图形的非超级边缘子集，并假设$ ne ne ne ne $ n ne $ν（e_i \ cup e_j $ j $ $ $ $然后，$ e = \ bigcup e_i $具有大小$ k $的彩虹匹配。此外，当$ e $是两部分图的边缘集时，边缘集$ e_i $的数量可以减少到$ 2K-1 $。
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 $\mathsf{NM}_k(G)$, which vanishes in all dimensions $d\geq 3k-4$, and all dimensions $d \geq 2k-3$ when $G$ is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes the 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 $ν(E_i\cup E_j)\geq k$ 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.
