Spectral radius and rainbow $k$-factors of graphs

TL;DR

This paper establishes a spectral condition involving the spectral radius for the existence of rainbow $k$-factors in a set of graphs sharing the same vertex set, extending understanding of graph factors through spectral graph theory.

Contribution

It provides a new spectral criterion for the existence of rainbow $k$-factors in graphs, generalizing previous results and characterizing extremal cases.

Findings

Spectral radius condition guarantees rainbow $k$-factors.

Identifies extremal graphs where the condition is tight.

Extends spectral graph theory to rainbow factor problems.

Abstract

Let $\mathcal{G}=\{G_1,\ldots, G_{\frac{kn}{2}}\}$ be a set of graphs on the same vertex set $V=\{1,\dots,n\}$ where $k\cdot n$ is even. We say $\mathcal{G}$ admits a rainbow $k$-factor if there exists a $k$-regular graph $F$ on the vertex set $V$ such that all edges of $F$ are from different members of $\mathcal{G}$. In this paper, we show a sufficient spectral condition for the existence of a rainbow $k$-factor for $k\geq 2$, which is that if $\rho(G_i)\geq\rho(K_{k-1}\vee(K_1\cup K_{n-k}))$ for each $G_i\in \mathcal{G}$, then $\mathcal{G}$ admits a rainbow $k$-factor unless $G_1=G_2=\cdots=G_{\frac{kn}{2}}\cong K_{k-1}\vee(K_1\cup K_{n-k})$.