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 [Degree conditions for the existence of a ₂,P₅\-factor in a graph, RAIRO: Oper. Res. { 57} (2023), 2231-2237] conjectured that a connected graph G with δ(G)≥ 5 contains a ₂,P₅\-factor if δ (G)≥ 3α(G)-1/4, 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 7K₃, where X is a spanning subgraph of K₃. Furthermore, we give two degree conditions for the existence of ₂, P₅\-factors, one of which is a stronger version of Wang's another conjecture. We also show the degree conditions are best possible.