### Abstract

We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence c_{k} → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cn^{ϵ}, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)^{1/2} is impossible in a maximal triangle‐free graph.)

Original language | English |
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Pages (from-to) | 11-24 |

Number of pages | 14 |

Journal | Journal of Graph Theory |

Volume | 18 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*18*(1), 11-24. https://doi.org/10.1002/jgt.3190180103

**Maximal triangle‐free graphs with restrictions on the degrees.** / Füredi, Z.; Seress, Ákos.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 18, no. 1, pp. 11-24. https://doi.org/10.1002/jgt.3190180103

}

TY - JOUR

T1 - Maximal triangle‐free graphs with restrictions on the degrees

AU - Füredi, Z.

AU - Seress, Ákos

PY - 1994

Y1 - 1994

N2 - We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)1/2 is impossible in a maximal triangle‐free graph.)

AB - We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)1/2 is impossible in a maximal triangle‐free graph.)

UR - http://www.scopus.com/inward/record.url?scp=0042190006&partnerID=8YFLogxK

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U2 - 10.1002/jgt.3190180103

DO - 10.1002/jgt.3190180103

M3 - Article

AN - SCOPUS:0042190006

VL - 18

SP - 11

EP - 24

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -