Regularity of powers of binomial edge ideals of complete multipartite graphs

Let G=Kn1,n2,…,nr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = {K_{{n_1},{n_2}, \ldots ,{n_r}}}$$\end{document} be a complete multipartite graph on [n] with n > r > 1 and JG being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity reg(JGt) is 2t +1 for any positive integer t.