### 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.

Original language | English |
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Pages (from-to) | 175-197 |

Number of pages | 23 |

Journal | Journal of Graph Theory |

Volume | 36 |

Issue number | 4 |

Publication status | Published - 2001 Apr |

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### Keywords

- Claw-free graphs
- Path factors
- Path packings

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*36*(4), 175-197.