The Signless Laplacian Spectral Radius and Rainbow Matchings (Hamilton Paths) of Graphs

Let and be integers satisfying 1≤m≤(n-2)/2, and let [n]={1,...,n}. Let be a family of graphs (not necessarily distinct) defined on the common vertex set . A graph H on V is said to be rainbow if there exists a bijection such that every edge satisfies In this paper, we prove that if the signless Laplacian radius satisfies the inequality For each , then contains a rainbow matching, except when all graphs in are identical and isomorphic to either or . Furthermore, for and , we show that iffor each , then admits a rainbow Hamilton path, unless all graphs in are identical and isomorphic to .