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

TL;DR

This paper establishes spectral radius conditions that guarantee the existence of k-factors in connected 1-tough graphs with minimum degree at least k, extending previous results on Hamilton cycles.

Contribution

It provides a complete solution to the problem of spectral radius conditions for the existence of k-factors in 1-tough graphs with minimum degree at least k.

Findings

Derived spectral radius bounds for k-factors in graphs

Extended known conditions from Hamilton cycles to k-factors

Solved an open problem in spectral graph theory

Abstract

A $k$-regular spanning subgraph of $G$ is called 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 $\delta(G)\ge k$? In this paper, we completely solve this problem.