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 language||English (US)|
|Number of pages||20|
|Journal||Journal of Graph Theory|
|State||Published - Sep 1996|
All Science Journal Classification (ASJC) codes
- Geometry and Topology