On k-Saturated Graphs with Restrictions on the Degrees

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 F₄(n, D) = 4n - 15 for n > n₀ 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.