Spectral radius and rainbow $k$-factors in a bipartite graph family

TL;DR

This paper establishes a spectral radius condition that guarantees the existence of a rainbow $k$-factor in a family of balanced bipartite graphs and characterizes the extremal graphs achieving this bound.

Contribution

It provides a tight spectral radius criterion for rainbow $k$-factors in bipartite graphs and characterizes the extremal graphs for this property.

Findings

Spectral radius condition guarantees rainbow $k$-factor existence.

Complete characterization of spectral extremal graphs.

Tight sufficient condition for bipartite graph families.

Abstract

Let $\mathcal{G}=\{G_1, G_2, \ldots , G_{kn}\}$ be a family of balanced bipartite graphs on the same vertex set $[2n]$. A rainbow $k$-factor of $\mathcal{G}$ is defined as a $k$-factor such that any two distinct edges come from different graphs in $\mathcal{G}.$ In this paper, we provide a tight sufficient condition in terms of the spectral radius for a family of balanced bipartite graphs $\mathcal{G}$ to contain a rainbow $k$-factor. Furthermore, we completely characterize the corresponding spectral extremal graph.