Spectral extremal problems on factors in tough graphs, and beyond

The toughness $\tau \left(G\right)=\min \{\frac{\left|S\right|}{c(G-S)}:S\text{is a cut set of vertices in}G\}$ for $G\ncong K_n$, which was initially proposed by Chvátal in 1973. A graph G is called t-tough if $\tau \left(G\right)\ge t$. 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). A natural and interesting problem arises: What is a tight spectral condition to guarantee the existence of factors among tough graphs?

A spanning k-tree of a connected graph G is a spanning tree with the degree of every vertex at most k, which is considered as a connected $[1,k]$-factor. We in this paper provide a tight sufficient condition based on the spectral radius for a connected $\frac{1}{k-\eta }$-tough graph to contain a spanning k-tree, where $k\ge 3$ is an integer and $\eta =\{0,1\}$.

Let $b\ge 1$ be an integer. An odd $[1,b]$-factor of a graph G is a spanning subgraph F such that for each $v\in V\left(G\right)$, $d_F\left(v\right)$ is odd and $1\le d_F\left(v\right)\le b$. We propose a tight sufficient condition in terms of the spectral radius for a connected $\frac{1}{b+1}$-tough graph to contain an odd $[1,b]$-factor. If $b=1$, an odd $[1,b]$-factor is called a 1-factor (perfect matching). We also present a tight sufficient condition in terms of the spectral radius for a connected $\frac{l}{l+1}$-tough graph to contain a 1-factor, where $l\ge 1$ is an integer.