Degree conditions for path-factors in graphs

A spanning subgraph H of a graph G is called a path-factor if every component of H is a path. Wang and Zhang [ RAIRO:RO 57 (2023) 2231–2237] conjectured that a connected graph G with δ ( G ) ≥ 5 contains a { P 2 , P 5 }-factor if δ ( G )≥3 α ( G )−14, where δ ( G ) and α ( G ) denote the minimum degree and independence number of G , respectively. We show that the conjecture is true except G ≅ X ∨ 7 K 3 , where X is a spanning subgraph of K 3 . Furthermore, we give two degree conditions for the existence of { P 2 , P 5 }-factors, one of which is a stronger version of Wang’s another conjecture. We also show the degree conditions are best possible.