Spectral radius and rainbow matchings of graphs

Let n, m be integers such that 1 <= m <= (n - 2)/2 and let [n] = {1, ..., n}. Let G = {G1, . . . , G(m+1)} be a family of graphs on the same vertex set [n]. In this paper, we prove that if for any i is an element of [m + 1], the spectral radius of G(i) is not less than max{2m, 1/2 (m -1 + root(m - 1)(2 )+ 4m(n - m))}, then G admits a rainbow matching, i.e. a choice of disjoint edges e(i) is an element of Gi, unless G(1) = G(2) = ... = G(m+1) and G(1) is an element of {K2m+1 boolean OR (n -2m - 1)K-1, K-m V (n - m)K-1}.(c) 2023 Elsevier Inc. All rights reserved.