| Abstract: | Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ 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 λ(G) = ?| V(G)|/3?; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ?(| V(G) | ? eb(G) + 2)/3 ?; and (t3) if G is claw‐free and Δ*‐free then λ(G) = ?| V(G) |/3? (here Δ* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n ? 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G ? S has a Λ‐factor for every S ? V(G) such that |S| = 3p and both V(G) ? S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001 |
