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Let k be an infinite field and S = k[x 1 ,. .. , x n ] the polynomial ring over k with each degx i = 1 and m = (x 1 ,. .. , x n) 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-Macaulayfatcat:hrslcxdyevcxrhtt24kfb2atl4