On packing 3-vertex paths in a graph

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