The Castelnuovo–Mumford regularity of binomial edge ideals

We prove a conjectured upper bound for the Castelnuovo–Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that reg(JG)≤n−1reg(JG)≤n−1 for any graph G with n vertices, which is not a path.