Signless Laplacian spectral conditions for rainbow matchings in a collection of bipartite graphs

Let ${\cal G}=\{G_1,\ldots,G_k\}$ be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition $(X,Y)$, where $|X|=a$, $|Y|=b$ and $2\le k\le a\le b$. A \emph{rainbow matching} of ${\cal G}$ is a set of pairwise disjoint edges that can be chosen from distinct members