Sufficient conditions for k-factors and spanning trees of graphs

For any integer k≥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 δ, which improves the sufficient condition of O (2021) based on the number of edges. For any integer k≥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∈V(T). 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 δ, where k≥1 is an integer.