### Abstract

A graph G is called C_{k}-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in C_{k}-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + 1 c_{1}n/k and n + c_{2}n/k for some positive constants c_{1} and c_{2}. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.

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

Pages (from-to) | 31-48 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 150 |

Issue number | 1-3 |

Publication status | Published - Apr 6 1996 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*150*(1-3), 31-48.

**Cycle-saturated graphs of minimum size.** / Barefoot, C. A.; Clark, L. H.; Entringer, R. C.; Porter, T. D.; Székely, L. A.; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 150, no. 1-3, pp. 31-48.

}

TY - JOUR

T1 - Cycle-saturated graphs of minimum size

AU - Barefoot, C. A.

AU - Clark, L. H.

AU - Entringer, R. C.

AU - Porter, T. D.

AU - Székely, L. A.

AU - Tuza, Z.

PY - 1996/4/6

Y1 - 1996/4/6

N2 - A graph G is called Ck-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in Ck-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + 1 c1n/k and n + c2n/k for some positive constants c1 and c2. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.

AB - A graph G is called Ck-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in Ck-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + 1 c1n/k and n + c2n/k for some positive constants c1 and c2. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.

UR - http://www.scopus.com/inward/record.url?scp=0041630480&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041630480&partnerID=8YFLogxK

M3 - Article

VL - 150

SP - 31

EP - 48

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -