### Abstract

Let G be a graph of order n satisfying d(u) + d(v) n for every edge uv of G. We show that the circumference - the length of a longest cycle - of G can be expressed in terms of a certain graph parameter, and can be computed in polynomial time. Moreover, we show that G contains cycles of every length between 3 and the circumference, unless G is complete bipartite. If G is 1-tough then it is pancyclic or G = Kr,r with r = n/2.

Original language | Undefined |
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Pages (from-to) | 253-256 |

Number of pages | 4 |

Journal | Journal of graph theory |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1997 |

### Keywords

- Circumference
- Closure
- Cycle
- Graph
- Degree
- IR-71423
- pancyclic
- sum
- tough
- METIS-140776
- Hamiltonian

## Cite this

Brandt, S., & Veldman, H. J. (1997). Degree sums for edges and cycle lengths in graphs.

*Journal of graph theory*,*25*(4), 253-256. https://doi.org/10.1002/(SICI)1097-0118(199708)25:4<253::AID-JGT2>3.0.CO;2-J