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

Hikoe Enomoto, Jun Fujisawa

Research output: Contribution to journalArticle

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{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.

Original languageEnglish
Pages (from-to)129-136
Number of pages8
JournalArs Combinatoria
Volume90
Publication statusPublished - 2009 Jan 1

Keywords

  • Fan-type condition
  • Long path

ASJC Scopus subject areas

  • Mathematics(all)

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