### Abstract

Let n (G) denote the number of vertices of a graph G and let α (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined byρ (G) = max {} fenced(frac(n (H), α (H)) : H ⊆ G),where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies ω (G) ≤ ρ (G) ≤ χ (G) where ω and χ denote the clique number and the chromatic number of G, respectively. We show that the interval [ω (G), ρ (G)] can be arbitrary large by estimating the Hall ratio of the Mycielski graphs.

Original language | English |
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Pages (from-to) | 1988-1990 |

Number of pages | 3 |

Journal | Discrete Mathematics |

Volume | 306 |

Issue number | 16 |

DOIs | |

Publication status | Published - Aug 28 2006 |

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### Keywords

- Fractional chromatic number
- Hall ratio
- Mycielski graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*306*(16), 1988-1990. https://doi.org/10.1016/j.disc.2005.09.020

**Hall ratio of the Mycielski graphs.** / Cropper, Mathew; Gyárfás, A.; Lehel, Jeno.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 306, no. 16, pp. 1988-1990. https://doi.org/10.1016/j.disc.2005.09.020

}

TY - JOUR

T1 - Hall ratio of the Mycielski graphs

AU - Cropper, Mathew

AU - Gyárfás, A.

AU - Lehel, Jeno

PY - 2006/8/28

Y1 - 2006/8/28

N2 - Let n (G) denote the number of vertices of a graph G and let α (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined byρ (G) = max {} fenced(frac(n (H), α (H)) : H ⊆ G),where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies ω (G) ≤ ρ (G) ≤ χ (G) where ω and χ denote the clique number and the chromatic number of G, respectively. We show that the interval [ω (G), ρ (G)] can be arbitrary large by estimating the Hall ratio of the Mycielski graphs.

AB - Let n (G) denote the number of vertices of a graph G and let α (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined byρ (G) = max {} fenced(frac(n (H), α (H)) : H ⊆ G),where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies ω (G) ≤ ρ (G) ≤ χ (G) where ω and χ denote the clique number and the chromatic number of G, respectively. We show that the interval [ω (G), ρ (G)] can be arbitrary large by estimating the Hall ratio of the Mycielski graphs.

KW - Fractional chromatic number

KW - Hall ratio

KW - Mycielski graphs

UR - http://www.scopus.com/inward/record.url?scp=33746766197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746766197&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2005.09.020

DO - 10.1016/j.disc.2005.09.020

M3 - Article

AN - SCOPUS:33746766197

VL - 306

SP - 1988

EP - 1990

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 16

ER -