### Abstract

A weighted graph is one in which every edge e is assigned a non-negative number, called the weight, of e. For a vertex v of a weighted graph, d ^{w}(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. Let G be a k-connected weighted graph where 2 ≤ k. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k independent vertices is at least m, (2) w(xz) = w(yz) for every vertex z ε N(x) ∩ N (y) with d(x,y) = 2, arid (4) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight.

Original language | English |
---|---|

Pages (from-to) | 225-232 |

Number of pages | 8 |

Journal | Ars Combinatoria |

Volume | 76 |

Publication status | Published - 2005 Jul |

### Fingerprint

### Keywords

- Heavy cycle
- Weighted degree sum
- Weighted graph

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{k}type condition for heavy cycles in weighted graphs.

*Ars Combinatoria*,

*76*, 225-232.

**A σ _{k} type condition for heavy cycles in weighted graphs.** / Enomoto, Hikoe; Fujisawa, Jun; Ota, Katsuhiro.

Research output: Contribution to journal › Article

_{k}type condition for heavy cycles in weighted graphs',

*Ars Combinatoria*, vol. 76, pp. 225-232.

_{k}type condition for heavy cycles in weighted graphs. Ars Combinatoria. 2005 Jul;76:225-232.

}

TY - JOUR

T1 - A σk type condition for heavy cycles in weighted graphs

AU - Enomoto, Hikoe

AU - Fujisawa, Jun

AU - Ota, Katsuhiro

PY - 2005/7

Y1 - 2005/7

N2 - A weighted graph is one in which every edge e is assigned a non-negative number, called the weight, of e. For a vertex v of a weighted graph, d w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. Let G be a k-connected weighted graph where 2 ≤ k. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k independent vertices is at least m, (2) w(xz) = w(yz) for every vertex z ε N(x) ∩ N (y) with d(x,y) = 2, arid (4) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight.

AB - A weighted graph is one in which every edge e is assigned a non-negative number, called the weight, of e. For a vertex v of a weighted graph, d w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. Let G be a k-connected weighted graph where 2 ≤ k. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k independent vertices is at least m, (2) w(xz) = w(yz) for every vertex z ε N(x) ∩ N (y) with d(x,y) = 2, arid (4) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight.

KW - Heavy cycle

KW - Weighted degree sum

KW - Weighted graph

UR - http://www.scopus.com/inward/record.url?scp=33644667582&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644667582&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33644667582

VL - 76

SP - 225

EP - 232

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -