Characterizations for P≥2P≥2-factor and P≥3P≥3-factor covered graphs

A P≥kP≥k-factor of a graph GG is a spanning subgraph FF of GG such that each component of FF is a path of order at least kk (k≥2k≥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≥2P≥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≥3P≥3-factor. We define the concept of a P≥kP≥k-factor covered graph, i.e. for each edge ee of GG, there is a P≥kP≥k-factor covering ee (k≥2k≥2). Based on these two results, we obtain respective necessary and sufficient conditions defining a P≥2P≥2-factor covered graph and a P≥3P≥3-factor covered graph.