On packing 3-vertex paths in a graph

Let G be a connected graph and let eb(G) and (G) denote the number of end-blocks and the maximum number of disjoint 3-vertex paths in G. We prove the following theorems on claw-free graphs: (t1) if G is claw-free and eb(G) 2 (and in particular, G is 2-connected) then (G)ˆbjV(G)j/3c; (t 2) if G is claw-free and eb(G) 2 then (G) b(jV(G)jÿ eb(G)‡ 2)/3c; and (t 3) if G is claw-free and -free then (G)ˆbjV(G)j/3c (here is a graph obtained from a triangle by attaching to each vertex a new dangling edge). We also give the following suf®cient condition for a graph to have a -factor: Let n and p be integers, 1 p nÿ 2, G a 2connected graph, and jV(G)j ˆ 3n. Suppose that G±S has a -factor for every S V(G) such that jSj ˆ 3p and both V(G)±S and S induce connected subgraphs in G. Then G has a -factor. ß 2001 John Wiley & Sons, Inc. J Graph Theory 36: