Path factors of bipartite graphs

A path on n vertices is denoted by P n . For any graph H , the number of isolated vertices of H is denoted by i(H) . Let G be a graph. A spanning subgraph F of G is called a { P 3 , P 4 , P 5 }‐factor of G if every component of F is one of P 3 , P 4 , and P 5 . In this paper, we prove that a bipartite graph G has a { P 3 , P 4 , P 5 }‐factor if and only if i(G − S − M) ≦ 2| S | + | M | for all S ⊆ V(G) and independent M ⊆ E(G) .