Cohen-Macaulay property of binomial edge ideals with girth of graphs

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. For a square-free monomial ideal I and a variable x, we give a condition under which I is Cohen-Macaulay if and only if ?I,x? is Cohen-Macaulay. Using this fact and computing the initial ideal, we prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only biconnected graphs with some whisker attached". As an application of our results, we solve some open problems. At the end, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than 5 or equal to infinity.