### Abstract

In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, α let f (k, α) be the maximum order of a graph G with independence number α (G) ≤ α, which has no k vertex-disjoint cycles. We prove that f (k, α) = 3 k + 2 α - 3 if 1 ≤ α ≤ 5 or 1 ≤ k ≤ 2, and f (k, α) ≥ 3 k + 2 α - 3 in general. We also prove the following results: (1) there exists a constant c_{α} (depending only on α) such that f (k, α) ≤ 3 k + c_{α}, (2) there exists a constant t_{k} (depending only on k) such that f (k, α) ≤ 2 α + t_{k}, and (3) there exists no absolute constant c such that f (k, α) ≤ c (k + α).

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

Pages (from-to) | 1493-1498 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 307 |

Issue number | 11-12 |

DOIs | |

Publication status | Published - 2007 May 28 |

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### Keywords

- Independence number
- Vertex-disjoint cycles

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*307*(11-12), 1493-1498. https://doi.org/10.1016/j.disc.2005.11.086

**Independence number and vertex-disjoint cycles.** / Egawa, Yoshimi; Enomoto, Hikoe; Jendrol, Stanislav; Ota, Katsuhiro; Schiermeyer, Ingo.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 307, no. 11-12, pp. 1493-1498. https://doi.org/10.1016/j.disc.2005.11.086

}

TY - JOUR

T1 - Independence number and vertex-disjoint cycles

AU - Egawa, Yoshimi

AU - Enomoto, Hikoe

AU - Jendrol, Stanislav

AU - Ota, Katsuhiro

AU - Schiermeyer, Ingo

PY - 2007/5/28

Y1 - 2007/5/28

N2 - In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, α let f (k, α) be the maximum order of a graph G with independence number α (G) ≤ α, which has no k vertex-disjoint cycles. We prove that f (k, α) = 3 k + 2 α - 3 if 1 ≤ α ≤ 5 or 1 ≤ k ≤ 2, and f (k, α) ≥ 3 k + 2 α - 3 in general. We also prove the following results: (1) there exists a constant cα (depending only on α) such that f (k, α) ≤ 3 k + cα, (2) there exists a constant tk (depending only on k) such that f (k, α) ≤ 2 α + tk, and (3) there exists no absolute constant c such that f (k, α) ≤ c (k + α).

AB - In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, α let f (k, α) be the maximum order of a graph G with independence number α (G) ≤ α, which has no k vertex-disjoint cycles. We prove that f (k, α) = 3 k + 2 α - 3 if 1 ≤ α ≤ 5 or 1 ≤ k ≤ 2, and f (k, α) ≥ 3 k + 2 α - 3 in general. We also prove the following results: (1) there exists a constant cα (depending only on α) such that f (k, α) ≤ 3 k + cα, (2) there exists a constant tk (depending only on k) such that f (k, α) ≤ 2 α + tk, and (3) there exists no absolute constant c such that f (k, α) ≤ c (k + α).

KW - Independence number

KW - Vertex-disjoint cycles

UR - http://www.scopus.com/inward/record.url?scp=33947243416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947243416&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2005.11.086

DO - 10.1016/j.disc.2005.11.086

M3 - Article

AN - SCOPUS:33947243416

VL - 307

SP - 1493

EP - 1498

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 11-12

ER -