Binomial edge ideals and bounds for their regularity

Let G be a simple graph on n vertices and JG denote the corresponding binomial edge ideal in S= K[x1, … , xn, y1, … , yn]. We prove that the Castelnuovo–Mumford regularity of JG is bounded above by c(G) + 1 , when G is a quasi-block graph or semi-block graph. We give another proof of Saeedi Madani–Kiani regularity upper bound conjecture for chordal graphs. We obtain the regularity of binomial edge ideals of Jahangir graphs. Later, we establish a sufficient condition for Hibi–Matsuda conjecture to be true.