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 less than or equal to D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D greater than or equal to (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 C-k -->, 0. The determination of K(c) is a finite problem on all intervals [gamma, infinity). For D = cn(8), 1/2 < epsilon < 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.) (C) 1994 John Wiley and Sons, Inc.