### 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 1 |

### Keywords

- Fan-type condition
- Long path

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Enomoto, H., & Fujisawa, J. (2009). Fan-type theorem for a long path passing through a specified vertex.

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