### Abstract

Let n_{1}, n_{2}, . . ., n_{k} be integers of at least two. Johansson gave a minimum degree condition for a graph of order exactly n_{1} + n_{2} + ⋯ + n_{k} to contain k vertex-disjoint paths of order n_{1}, n_{2}, . . ., n_{k}, respectively. In this paper, we extend Johansson's result to a corresponding packing problem as follows. Let G be a connected graph of order at least n_{1} + n_{2} + ⋯ + n_{k}. Under this notation, we show that if the minimum degree sum of three independent vertices in G is at least (formula presented) then G contains k vertex-disjoint paths of order n_{1}, n_{2}, . . ., n_{k}, respectively, or else n_{1} = n_{2} = ⋯ = n_{k} = 3, or k = 2 and n_{1} = n_{2} = odd. The graphs in the exceptional cases are completely characterized. In particular, these graphs have more than n_{1} + n_{2} + ⋯ + n_{k} vertices.

Original language | English |
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Pages (from-to) | 23-31 |

Number of pages | 9 |

Journal | Ars Combinatoria |

Volume | 61 |

Publication status | Published - 2001 Dec 1 |

### ASJC Scopus subject areas

- Mathematics(all)