2023/08/26 by Guoyan Ao, Ruifang Liu, Ao, Guoyan +7
Mathematics · Engineering · #Graph theory and applications #graph theory and CDMA systems #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2308.13874
For any integer k≥1, a graph G has a k-factor if it contains a k-regular spanning subgraph. In this paper we prove 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 \citeO2021 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 \citeLi2016, we present a tight spectral condition for an m-connected graph to have a spanning k-tree, which extends the result of Fan, Goryainov, Huang and Lin \citeFan2021 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.