Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral radius, toughness, binding number and $k$-factor of graphs

Yuanyuan Chen, Huiqiu Lin, Shucheng Li

Abstract

A $k$-regular spanning subgraph of $G$ is called a $k$-factor. In this paper, we provide spectral radius and edge conditions to ensure that a graph $G$ with $δ(G)\ge k$ admits a $k$-factor. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] presented a tight sufficient condition in terms of the spectral radius for a connected 1-tough graph to contain a connected 2-factor (Hamilton cycle). Then it is interesting to consider the following problem: What is the spectral radius condition to guarantee the existence of a $k$-factor with $k\ge3$ in a connected 1-tough graph $G$ with $δ(G)\ge k$? We completely solve this problem, and we further obtain a sufficient spectral radius condition for the existence of a $k$-factor in a connected 1-binding graph, which solves an important problem posed by Fan and Lin [Electron. J. Combin. 31 (2024) 1--30].

Spectral radius, toughness, binding number and $k$-factor of graphs

Abstract

A -regular spanning subgraph of is called a -factor. In this paper, we provide spectral radius and edge conditions to ensure that a graph with admits a -factor. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] presented a tight sufficient condition in terms of the spectral radius for a connected 1-tough graph to contain a connected 2-factor (Hamilton cycle). Then it is interesting to consider the following problem: What is the spectral radius condition to guarantee the existence of a -factor with in a connected 1-tough graph with ? We completely solve this problem, and we further obtain a sufficient spectral radius condition for the existence of a -factor in a connected 1-binding graph, which solves an important problem posed by Fan and Lin [Electron. J. Combin. 31 (2024) 1--30].
Paper Structure (3 sections, 11 theorems, 35 equations, 1 figure)

This paper contains 3 sections, 11 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorems \ref{['ta']}-\ref{['tc']}

Key Result

Theorem 1

Let $kn$ be an even integer and $k\ge1$. If $G$ is a graph of order $n\ge 4k-1$ with $\rho(G)>\rho(K_{k-1}\vee(K_{n-k}\cup K_1))$, then $G$ contains a $k$-factor.

Figures (1)

  • Figure :

Theorems & Definitions (15)

  • Conjecture 1: Cho, Hyun, O and Park E. Cho
  • Theorem 1: Fan and Lin D. Fan3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: Fan and Lin D. Fan1
  • Corollary 1
  • Lemma 2.1: See Nosal
  • Lemma 2.2
  • Lemma 2.3: See Y. Hong1V. Nikiforov1
  • ...and 5 more