Characterizing P≥2-factor and P≥2-factor covered graphs with respect to the size or the spectral radius

A P->= k-factor (k >= 2) of a graph G is a spanning subgraph of G in which each component is a path of order at least k. A graph G is called a P->= k-factor covered graph if for each edge e of G, there is a P->= k-factor covering e. In this paper, we first establish two lower bounds on the size of a graph G, in which one bound guarantees that G contains a P->= 2-factor, the other bound ensures that the graph G is a P->= 2-factor covered graph. Then we establish two lower bounds on the spectral radius of a graph G, in which one bound guarantees that the graph G has a P->= 2-factor, the other bound ensures that the graph G is a P->= 2-factor covered graph. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible. (C) 2021 Elsevier B.V. All rights reserved.