### Abstract

A graph G is called k-saturated, where k ≥ 3 is an integer, if G is K^{k}-free but the addition of any edge produces a K^{k} (we denote by K^{k} a complete graph on k vertices). We investigate k-saturated graphs, and in particular the function F_{k}(n, D) defined as the minimal number of edges in a k-saturated graph on n vertices having maximal degree at most D. This investigation was suggested by Hajnal, and the case k = 3 was studied by Füredi and Seress. The following are some of our results. For k = 4, we prove that F_{4}(n, D) = 4n - 15 for n > n_{0} and [(2n - 1)/3] ≤ D ≤ n - 2. For arbitrary k, we show that the limit lim_{n→∞} F_{k}(n, cn)/n exists for all 0 <c ≤ 1, except maybe for some values of c contained in a sequence c_{i} → 0. We also determine the asymptotic behavior of this limit for c → 0. We construct, for all k and all sufficiently large n, a k-saturated graph on n vertices with maximal degree at most 2k√n, significantly improving an upper bound due to Hanson and Seyffarth.

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

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Graph Theory |

Volume | 23 |

Issue number | 1 |

Publication status | Published - Sep 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*23*(1), 1-20.

**On k-Saturated Graphs with Restrictions on the Degrees.** / Alon, Noga; Erdős, P.; Holzman, Ron; Krivelevich, Michael.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 23, no. 1, pp. 1-20.

}

TY - JOUR

T1 - On k-Saturated Graphs with Restrictions on the Degrees

AU - Alon, Noga

AU - Erdős, P.

AU - Holzman, Ron

AU - Krivelevich, Michael

PY - 1996/9

Y1 - 1996/9

N2 - A graph G is called k-saturated, where k ≥ 3 is an integer, if G is Kk-free but the addition of any edge produces a Kk (we denote by Kk a complete graph on k vertices). We investigate k-saturated graphs, and in particular the function Fk(n, D) defined as the minimal number of edges in a k-saturated graph on n vertices having maximal degree at most D. This investigation was suggested by Hajnal, and the case k = 3 was studied by Füredi and Seress. The following are some of our results. For k = 4, we prove that F4(n, D) = 4n - 15 for n > n0 and [(2n - 1)/3] ≤ D ≤ n - 2. For arbitrary k, we show that the limit limn→∞ Fk(n, cn)/n exists for all 0 i → 0. We also determine the asymptotic behavior of this limit for c → 0. We construct, for all k and all sufficiently large n, a k-saturated graph on n vertices with maximal degree at most 2k√n, significantly improving an upper bound due to Hanson and Seyffarth.

AB - A graph G is called k-saturated, where k ≥ 3 is an integer, if G is Kk-free but the addition of any edge produces a Kk (we denote by Kk a complete graph on k vertices). We investigate k-saturated graphs, and in particular the function Fk(n, D) defined as the minimal number of edges in a k-saturated graph on n vertices having maximal degree at most D. This investigation was suggested by Hajnal, and the case k = 3 was studied by Füredi and Seress. The following are some of our results. For k = 4, we prove that F4(n, D) = 4n - 15 for n > n0 and [(2n - 1)/3] ≤ D ≤ n - 2. For arbitrary k, we show that the limit limn→∞ Fk(n, cn)/n exists for all 0 i → 0. We also determine the asymptotic behavior of this limit for c → 0. We construct, for all k and all sufficiently large n, a k-saturated graph on n vertices with maximal degree at most 2k√n, significantly improving an upper bound due to Hanson and Seyffarth.

UR - http://www.scopus.com/inward/record.url?scp=1842429967&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1842429967&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:1842429967

VL - 23

SP - 1

EP - 20

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -