### Abstract

Let G be a graph, and let p(G) and c(G) be the order of a longest path and a longest cycle of G, respectively. In [J. Graph Theory 30 (1999), 91-99], Saito proved that if G is a 2-connected graph with p(G) - c(G) ≥ 2, then p(G) ≥ σ _{3}(G) - 1. In this paper, we evaluate the length of a longest path of G by using σ _{4}(G). Specifically, our main results are the following. (i) If G is a 3-connected graph with p(G) - c(G) ≥ 3, then p(G) ≥ σ _{4}(G)-5, and (ii) if G is a 3-connected graph withp(G)-c(G) ≥ 2, then p(G) ≥ 3σ _{4}(G)/4-1. The statement (ii) is a generalization of Saito's theorem for 3-connected graphs. In fact, we characterize all graphs G with p(G) - c(G) ≥ 2 and p(G) = 3σ _{4}(G)/4-1. Following these results, we propose a conjecture, and obtain an application for the problem concerning the existence of vertex-disjoint paths.

Original language | English |
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Pages (from-to) | 91-107 |

Number of pages | 17 |

Journal | Australasian Journal of Combinatorics |

Volume | 47 |

Publication status | Published - 2010 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*47*, 91-107.