Sufficient conditions for graphs with {<i>P</i>₂, <i>P</i>₅}-factors

Abstract

For a graph G , a spanning subgraph F of G is called an { P 2 , P 5 }-factor if every component of F is isomorphic to P 2 or P 5 , where P k denotes the path of order k . It was proved by Egawa and Furuya that if G satisfies 3 c 1 ( G − S ) + 2 c 3 ( G − S ) ≤ 4| S | + 1 for all S ⊆ V ( G ), then G has a { P 2 , P 5 }-factor, where c k ( G − S ) denotes the number of components of G − S with order k . By this result, we give some other sufficient conditions for a graph to have a { P 2 , P 5 }-factor by various graphic parameters such as toughness, binding number, degree sums, etc. Moreover, we obtain some regular graphs and some K (1, r ) -free graphs having { P 2 , P 5 }-factors.