The Existence of a Path-Factor without Small Odd Paths

A {P2, P5}-factor of a graph is a spanning subgraph of the graph each of whose components is isomorphic to either P2 or P5, where Pn denote the path of order n. In this paper, we show that if a graph G satisfies c1(G−X)+ 2 3c3(G−X) 6 4 3 |X|+ 1 3 for all X ⊆ V (G), then G has a {P2, P5}-factor, where ci(G − X) is the number of components C of G − X with |V (C)| = i. Moreover, it is shown that above condition is sharp.