### 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 |
---|---|

Pages (from-to) | 175-197 |

Number of pages | 23 |

Journal | Journal of Graph Theory |

Volume | 36 |

Issue number | 4 |

Publication status | Published - 2001 Apr |

### Keywords

- Claw-free graphs
- Path factors
- Path packings

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*36*(4), 175-197.

**On packing 3-vertex paths in a graph.** / Kaneko, Atsushi; Kelmans, Alexander; Nishimura, Tsuyoshi.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 36, no. 4, pp. 175-197.

}

TY - JOUR

T1 - On packing 3-vertex paths in a graph

AU - Kaneko, Atsushi

AU - Kelmans, Alexander

AU - Nishimura, Tsuyoshi

PY - 2001/4

Y1 - 2001/4

N2 - 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.

AB - 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.

KW - Claw-free graphs

KW - Path factors

KW - Path packings

UR - http://www.scopus.com/inward/record.url?scp=31244438088&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=31244438088&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:31244438088

VL - 36

SP - 175

EP - 197

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 4

ER -