IMPROVED BOUNDS FOR THE REGULARITY OF POWERS OF EDGE IDEALS OF GRAPHS

Let G be a graph with edge ideal I (G). We recall the notions, given by Seyed Fakhari and Yassemi (2018), of min-match{K2,C5}(G) and ind-match{K2,C5}(G). We show that reg(I(G)s) & LE; 2s +min-match{K2,C5}(G) - 1, for all s & GE; 1, which implies that reg(I (G)s) & LE; 2s +min-match(G) - 1. Moreover, we show that reg(I(G)s) & GE; 2s +ind-match{K2,C5}(G) - 2, and if ind-match{K2,C5}(G) is an odd integer, then reg(I(G)s) & GE; 2s + ind-match{K2,C5}(G) - 1. As a consequence, we estimate the regularity of powers of edge ideals of Cohen-Macaulay graphs with girth(G) & GE; 5. Furthermore, it is shown that reg(I (G)s) & LE; 2s + ord-match(G) - 1, where ord-match(G) denotes the ordered matching number of G.