Maximal triangle-free graphs with restrictions on the degrees

We investigate the problem that at least how many edges must a maximal triangle-free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n - 1)1/2 is impossible in a maximal triangle-free graph.)