## 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 (US) |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Graph Theory |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1996 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Geometry and Topology