Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs

Let G be a finite simple graph on n vertices and JG denote the corresponding binomial edge ideal in S=K[x1,…,xn,y1,…,yn]. We compute the Castelnuovo-Mumford regularity of S/JG when JG is the binomial edge ideal of a Cohen-Macaulay bipartite graph. We achieve this by computing the regularity of certain bipartite subgraphs and some intermediate graphs, called k-fan graphs. In this process, we also obtain a class of graphs which satisfy the regularity conjecture of Saeedi Madani and Kiani.