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

Pages (from-to) | 23-31 |

Number of pages | 9 |

Journal | Ars Combinatoria |

Volume | 61 |

Publication status | Published - 2001 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Vertex-Disjoint Paths in Graphs.** / Egawa, Yoshimi; Ota, Katsuhiro.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Vertex-Disjoint Paths in Graphs

AU - Egawa, Yoshimi

AU - Ota, Katsuhiro

PY - 2001

Y1 - 2001

N2 - Let n1, n2, . . ., nk be integers of at least two. Johansson gave a minimum degree condition for a graph of order exactly n1 + n2 + ⋯ + nk to contain k vertex-disjoint paths of order n1, n2, . . ., nk, 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 n1 + n2 + ⋯ + nk. 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 n1, n2, . . ., nk, respectively, or else n1 = n2 = ⋯ = nk = 3, or k = 2 and n1 = n2 = odd. The graphs in the exceptional cases are completely characterized. In particular, these graphs have more than n1 + n2 + ⋯ + nk vertices.

AB - Let n1, n2, . . ., nk be integers of at least two. Johansson gave a minimum degree condition for a graph of order exactly n1 + n2 + ⋯ + nk to contain k vertex-disjoint paths of order n1, n2, . . ., nk, 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 n1 + n2 + ⋯ + nk. 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 n1, n2, . . ., nk, respectively, or else n1 = n2 = ⋯ = nk = 3, or k = 2 and n1 = n2 = odd. The graphs in the exceptional cases are completely characterized. In particular, these graphs have more than n1 + n2 + ⋯ + nk vertices.

UR - http://www.scopus.com/inward/record.url?scp=0346977785&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346977785&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0346977785

VL - 61

SP - 23

EP - 31

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -