Topology of complexes of graphs with bounded matching number

Abstract

Given a graph $G$ on the vertex set $V$, the non-matching complex of $G$, $NM_{k}(G)$, is the family of subgraphs $G'\subset G$ whose matching number $\nu(G')$ is strictly less than $k$. As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of $NM_{k}(K_{n})$ and $NM_{k}(K_{r,s})$ to arbitrary graphs G, we show that (i) $NM_{k}(G)$ is $(3k − 3)$-Leray, and (ii) if $G$ is bipartite, then $NM_{k}(G)$ is $(2k − 2)$-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex $NM_{k}(G)$, which vanishes in all dimensions $d \geq 3k − 4$, and all dimensions $d \geq 2k − 3$ when $G$ is bipartite. Following a similar line of the proof, we answer a question raised by Linusson et al. on the homotopy type of $NM_{k}(G)$ when $G$ is a complete multipartite graph with a vertex class consisting of a single vertex. As a corollary, we have the following rainbow matching theorem which generalizes the result by Aharoni et. al. and Drisko’s theorem: Let $E_1 , . . . , E_{3k−2}$ be non-empty edge subsets of a graph and suppose that $\nu(E_{i} \cup E_j) \geq k$ for every $i \neq j$. 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. We also consider a related collaborative rainbow matching problem, and construct examples showing that certain topological methods do not seem to be applicable.