### Abstract

Let G be a 2-connected graph with maximum degree Δ(G) ≥ d, x and z be distinct vertices of G, and W be a subset of V (G)\{x, z} such that |W| ≤ d - 1. Hirohata proved that if max{d_{G}(u), d_{G}(v)} ≥ d for every pair of vertices u, v ∈ V(G)\({x,z} ∪ W) such that d _{G}(u, v) = 2, then x and z are joined by a path of length at least d - |W|. In this paper, we show that if G satisfies the conditions of Hirohata's theorem, then for any given vertex y such that d_{G}(y) ≥ d, x and z are joined by a path of length at least d - |W| which contains y.

Original language | English |
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Pages (from-to) | 129-136 |

Number of pages | 8 |

Journal | Ars Combinatoria |

Volume | 90 |

Publication status | Published - 2009 Jan |

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### Keywords

- Fan-type condition
- Long path

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Ars Combinatoria*,

*90*, 129-136.

**Fan-type theorem for a long path passing through a specified vertex.** / Enomoto, Hikoe; Fujisawa, Jun.

Research output: Contribution to journal › Article

*Ars Combinatoria*, vol. 90, pp. 129-136.

}

TY - JOUR

T1 - Fan-type theorem for a long path passing through a specified vertex

AU - Enomoto, Hikoe

AU - Fujisawa, Jun

PY - 2009/1

Y1 - 2009/1

N2 - Let G be a 2-connected graph with maximum degree Δ(G) ≥ d, x and z be distinct vertices of G, and W be a subset of V (G)\{x, z} such that |W| ≤ d - 1. Hirohata proved that if max{dG(u), dG(v)} ≥ d for every pair of vertices u, v ∈ V(G)\({x,z} ∪ W) such that d G(u, v) = 2, then x and z are joined by a path of length at least d - |W|. In this paper, we show that if G satisfies the conditions of Hirohata's theorem, then for any given vertex y such that dG(y) ≥ d, x and z are joined by a path of length at least d - |W| which contains y.

AB - Let G be a 2-connected graph with maximum degree Δ(G) ≥ d, x and z be distinct vertices of G, and W be a subset of V (G)\{x, z} such that |W| ≤ d - 1. Hirohata proved that if max{dG(u), dG(v)} ≥ d for every pair of vertices u, v ∈ V(G)\({x,z} ∪ W) such that d G(u, v) = 2, then x and z are joined by a path of length at least d - |W|. In this paper, we show that if G satisfies the conditions of Hirohata's theorem, then for any given vertex y such that dG(y) ≥ d, x and z are joined by a path of length at least d - |W| which contains y.

KW - Fan-type condition

KW - Long path

UR - http://www.scopus.com/inward/record.url?scp=60749116849&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60749116849&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:60749116849

VL - 90

SP - 129

EP - 136

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -