Spectral radius and rainbow matchings of graphs

TL;DR

This paper establishes spectral radius conditions ensuring the existence of rainbow matchings in a family of graphs on the same vertex set, with specific exceptions characterized by particular graph structures.

Contribution

It provides new spectral radius criteria for rainbow matchings in graphs, extending previous combinatorial results with precise spectral bounds and characterizing extremal cases.

Findings

Spectral radius thresholds guarantee rainbow matchings.

Identifies specific extremal graph structures where conditions fail.

Extends combinatorial matching results to spectral graph theory.

Abstract

Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ 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 $G_1=G_2=\ldots=G_{m+1}$ and $G_1\in \{K_{2m+1}\cup (n-2m-1)K_1, K_m\vee (n-m)K_1\}$.