Gotzmann Ideals and applications to graphs

Let k be an infinite field and S = k[x1, . . . , xn] the polynomial ring over k with each degxi = 1 and m = (x1 , . . . , xn ) the graded maximal ideal of S. A graded ideal I generated in degree d is called a Gotzmann ideal if the number of generators of mI is the smallest possible, namely, equal to the number of generators of (mI)lex. A graph G is called Gotzmann if the edge ideal I(G) is a Gotzmann ideal. We determine some classes of Gotzmann graphs and we characterize all Cohen-Macaulay graphs which are Gotzmann and principal Borel.