On k -saturated graphs with restrictions on the degrees

TLDR: In this article, it was shown 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.

Abstract: A graph G is called k-saturated, where k ≥ 3 is an integer, if G is K-free but the addition of any edge produces a K (we denote by K 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 Furedi 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 behaviour 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. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by the Fund for Basic Research administered by the Israel Academy of Sciences. †Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary, and Department of Mathematics, Technion Israel Institute of Technology, Haifa, Israel. ‡Department of Mathematics, Technion Israel Institute of Technology, Haifa, Israel. Research supported by the Tragovnik research fund and by the fund for the promotion of research at the Technion. Corresponding author. §Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by a Charles Clore Fellowship.