Spectral conditions for rainbow matchings of bipartite graphs

Abstract

Let $\mathcal{G}=\{G_1,G_2,\cdots,G_k\}$ be a family of subgraphs of complete bipartite graph $K_{a,b}$ for $k\leq a\leq b$. In this paper, we prove that, if the spectral radius $\lambda(G_i)$ of $G_{i}$ satisfies $\lambda(G_i)\geq \sqrt{(k-1)b}$ for all $i\in [k]$, then $\mathcal{G}$ contains a rainbow matching unless $G_1=G_2=\cdots=G_k \cong K_{k-1,b}\cup \overline{K_{a-k+1}}$.