Given a graph on the vertex set , the non-matching complex of , , is the family of subgraphs whose matching number is strictly less than . As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of and to arbitrary graphs , we show that (i) is -Leray, and (ii) if is bipartite, then is -Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex , which vanishes in all dimensions , and all dimensions when is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes the result by Aharoni et. al. and Drisko’s theorem: Let be non-empty edge subsets of a graph and suppose that for every . Then has a rainbow matching of size . Furthermore, the number of edge sets can be reduced to when is the edge set of a bipartite graph.
This is a joint work with Andreas Holmsen.