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

Source abstract

Let ${\cal G}=\{{G_1},\ldots,{G_k}\}$ be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition $(X,Y)$ with $|X|=a$ and $|Y|=b$, where $k$, $a$ and $b$ are positive integers. A collection ${\cal G}$ of bipartite graphs admits a rainbow matching if there exists a set of pairwise disjoint edges such that any two edges are from distinct bipartite graphs of ${\cal G}$. Denote by $q({G})$ the signless Laplacian spectral radius of a bipartite graph $G$. In this paper, we prove that if $q({G_i})\ge b+k-1+\sqrt{(k-1)b}$ for each ${G_i}\in{\cal G}=\{{G_1}, \ldots, {G_k}\}$, where $2\leq k\leq a\leq b$, then ${\cal G}$ admits a rainbow matching of size $k$ unless ${G_1}={G_2}=\cdots={G_k}\cong{K_{k-1,b}}\cup\overline{K_{a-k+1}}$, by using the shifting technique.