Spectral radius and rainbow matchings of graphs

Abstract

Let $n,m$ be integers such that $1\le m\le (n-2)/2$ and let $[n]=\{1,\dots ,n\}$. Let $\mathcal{G}=\{G_1,\dots ,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the spectral radius of $G_i$ is not less than $\max \{2m,\frac{1}{2}(m-1+\sqrt{(m-1{)}^2+4m(n-m)})\}$, then $\mathcal{G}$ admits a rainbow matching, i.e. a choice of disjoint edges $e_i\in G_i$, unless $G_1=G_2=\dots =G_{m+1}$ and $G_1\in \{K_{2m+1}\cup (n-2m-1)K_1,K_m\vee (n-m)K_1\}$.