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

Meng Chen, Ruifang Liu

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.

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

Abstract

Let be a family of balanced bipartite graphs on the same vertex set . A rainbow -factor of is defined as a -factor such that any two distinct edges come from different graphs in In this paper, we provide a tight sufficient condition in terms of the spectral radius for a family of balanced bipartite graphs to contain a rainbow -factor. Furthermore, we completely characterize the corresponding spectral extremal graph.
Paper Structure (3 sections, 12 theorems, 25 equations)

This paper contains 3 sections, 12 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1.2']}

Key Result

Theorem 1.1

Let $2\leq k\leq \frac{n}{2}-1,$ and let $G$ be a connected balanced bipartite graph with order $n.$ If then $G$ contains a $k$-factor unless $G\cong K_{k-1, n-1}\sqcup \widehat{K_{n-k+1, 1}}.$

Theorems & Definitions (27)

  • Theorem 1.1: Fan and LinFan2026
  • Theorem 1.2: Shi, Li and ChenShi2024
  • Corollary 1.1: Shi, Li and ChenShi2024
  • Theorem 1.3
  • Lemma 2.1: Csikváricsikvari2009conjecture
  • Lemma 2.2: Guo et al.guo2023spectral
  • Lemma 2.3: Zhang and zhangzhangzhang2025
  • Lemma 2.4: Brouwer and Haemersbrouwer2, Godsil and RoyleGodsil
  • Lemma 3.1
  • proof
  • ...and 17 more