Characterizations for -factor and -factor covered graphs

A P ≥ k -factor of a graph G is a spanning subgraph F of G such that each component of F is a path of order at least k ( k ≥ 2 ). Akiyama et al. [J. Akiyama, D. Avis, H. Era, On a {1, 2}-factor of a graph, TRU Math. 16 (1980) 97–102] obtained a necessary and sufficient condition for a graph with a P ≥ 2 -factor. Kaneko [A. Kaneko, A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two, J. Combin. Theory Ser. B 88 (2003) 195–218] gave a characterization of a graph with a P ≥ 3 -factor. We define the concept of a P ≥ k -factor covered graph, i.e. for each edge e of G , there is a P ≥ k -factor covering e ( k ≥ 2 ). Based on these two results, we obtain respective necessary and sufficient conditions defining a P ≥ 2 -factor covered graph and a P ≥ 3 -factor covered graph.