Sufficient conditions for spanning $k$-trees in tough graphs

The toughness of a graph $G$, denoted by $\tau(G)$, is defined by $\tau(G)=$min $\{\frac{|S|}{c(G-S)}:S\subseteq V(G)$ and $c(G-S)\geq2\}$. A graph $G$ is said to be $\tau$-tough if $\tau(G)\geq \tau$. Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_{T}(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. A $k$-tree $T$ is a spanning $k$-tree if $T$ is a spanning subgraph of a connected graph $G$. In 1989, Win [Graphs Combin. 5 (1989) 201--205] proved that if $\tau(G)\geq\frac{1}{k-2}$, where $k\geq3$, then $G$ contains a spanning $k$-tree. Liu, Fan and Shu [Discrete Math. 348 (2025) 114593] provided a tight sufficient condition based on the spectral condition for connected $\frac{1}{k}$-tough and $\frac{1}{k-1}$-tough graphs to contain a spanning $k$-tree, where $k\geq3$ is an integer. A natural and interesting problem arises: Can the value of $\tau$ be refined? When $\frac{1}{k-2}>\tau\geq\frac{1}{k-1}$, we initially establish a lower bound on the size to ensure that a connected $\frac{t}{t(k-2)+1}$-tough graph $G$ contains a spanning $k$-tree, where $k\geq3$ and $t\geq1$ are integers. Meanwhile, we provide two sufficient conditions in terms of spectral radius and signless Laplacian spectral radius for a connected $\frac{t}{t(k-2)+1}$-tough graph $G$ to contain a spanning $k$-tree, where $k\geq3$ and $t\geq1$ are integers. When $t=1$, we obtain the result $\eta=1$ from Liu, Fan and Shu.