抄録
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.
| 元の言語 | English |
|---|---|
| ページ（範囲） | 91-107 |
| ページ数 | 17 |
| ジャーナル | Australasian Journal of Combinatorics |
| 巻 | 47 |
| 出版物ステータス | Published - 2010 |
| 外部発表 | Yes |
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
これを引用
Relationships between the length of a longest path and the relative length. / Chiba, Shuya; Matsubara, Ryota; Tsugaki, Masao.
：: Australasian Journal of Combinatorics, 巻 47, 2010, p. 91-107.研究成果: Article
}
TY - JOUR
T1 - Relationships between the length of a longest path and the relative length
AU - Chiba, Shuya
AU - Matsubara, Ryota
AU - Tsugaki, Masao
PY - 2010
Y1 - 2010
N2 - 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.
AB - 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=77953149107&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:77953149107
VL - 47
SP - 91
EP - 107
JO - Australasian Journal of Combinatorics
JF - Australasian Journal of Combinatorics
SN - 1034-4942
ER -