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

Abstract

Let $\mathcal{G}=\{G_1,\dots ,G_k\}$ 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 $\mathcal{G}$ is a set of pairwise disjoint edges that can be chosen from distinct members of $\mathcal{G}$. Denote by $q(G)$ the signless Laplacian spectral radius of a graph $G$. In this paper, we prove that if $q(G_i)\ge b+k-1$ for each $i\in \{1,2,\dots ,k\}$, then $\mathcal{G}$ admits a rainbow matching of size $k$ unless $G_1=\cdots =G_k\cong K_{k-1,b}\cup \overline{K_{a-k+1}}$, and show that the threshold is sharp and attained by the exceptional collection. The condition is also extended to larger collections for a prescribed level $t$. In addition, we obtain a lower bound for the rainbow matching number in terms of the ordered signless Laplacian spectral radii of the members, and provide a stability version of the extremal characterization. In the proofs, we use the shifting technique and a quotient matrix arising from an equitable partition of a signless Laplacian matrix.