Sufficient conditions for k-factors and spanning trees of graphs

For any integer $k\ge 1,$ a graph $G$ has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper, we present a sufficient condition in terms of the number of $r$-cliques to guarantee the existence of a $k$-factor in a graph with minimum degree at least $\delta$, which improves the sufficient condition of O (2021) based on the number of edges. For any integer $k\ge 2,$ a spanning $k$-tree of a connected graph $G$ is a spanning tree in which every vertex has degree at most $k$. Motivated by the technique of Li and Ning (2016), we present a tight spectral condition for an $m$-connected graph to have a spanning $k$-tree, which extends the result of Fan et al. (2022) from $m=1$ to general $m$. Let $T$ be a spanning tree of a connected graph. The leaf degree of $T$ is the maximum number of leaves adjacent to $v$ in $T$ for any $v\in V\left(T\right)$. We provide a tight spectral condition for the existence of a spanning tree with leaf degree at most $k$ in a connected graph with minimum degree $\delta$, where $k\ge 1$ is an integer.