 
Summary: On ksaturated graphs with restrictions on the degrees
Noga Alon
Paul Erdos
Ron Holzman
Michael Krivelevich§
February 22, 2002
Abstract
A graph G is called ksaturated, 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 ksaturated graphs, and in particular the function
Fk(n, D) defined as the minimal number of edges in a ksaturated graph on n
vertices having maximal degree at most D. This investigation was suggested by
Hajnal, and the case k = 3 was studied by F¨uredi and Seress. The following are
some of our results. For k = 4, we prove that F4(n, D) = 4n  15 for n > n0 and
2n1
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
