On k-Saturated Graphs with Restrictions on the Degrees

Noga Alon, P. Erdős, Ron Holzman, Michael Krivelevich

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

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 <c ≤ 1, except maybe for some values of c contained in a sequence ci → 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 languageEnglish
Pages (from-to)1-20
Number of pages20
JournalJournal of Graph Theory
Volume23
Issue number1
Publication statusPublished - Sep 1996

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Restriction
Graph in graph theory
Complete Graph
Asymptotic Behavior
Upper bound
Denote
Integer
Arbitrary

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Alon, N., Erdős, P., Holzman, R., & Krivelevich, M. (1996). On k-Saturated Graphs with Restrictions on the Degrees. 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.

In: Journal of Graph Theory, Vol. 23, No. 1, 09.1996, p. 1-20.

Research output: Contribution to journalArticle

Alon, N, Erdős, P, Holzman, R & Krivelevich, M 1996, 'On k-Saturated Graphs with Restrictions on the Degrees', Journal of Graph Theory, vol. 23, no. 1, pp. 1-20.
Alon N, Erdős P, Holzman R, Krivelevich M. On k-Saturated Graphs with Restrictions on the Degrees. Journal of Graph Theory. 1996 Sep;23(1):1-20.
Alon, Noga ; Erdős, P. ; Holzman, Ron ; Krivelevich, Michael. / On k-Saturated Graphs with Restrictions on the Degrees. In: Journal of Graph Theory. 1996 ; Vol. 23, No. 1. pp. 1-20.
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