ON THE REGULARITY OF SMALL SYMBOLIC POWERS OF EDGE IDEALS OF GRAPHS

Assume that G is a graph with edge ideal I (G) and let I (G)((s)) denote the s-th symbolic power of I (G). It is proved that for every integer s >= 1, reg(I(G)((s+1))) <= max {reg(I(G)) + 2s, reg (I(G)((s+1)) + I(G)(s))} As a consequence, we conclude that reg( I (G)((2))) <= reg( I (G)) + 2, and reg( I (G)((3))) <= reg( I (G))+ 4. Moreover, it is shown that if for some integer k >= 1, the graph G has no odd cycle of length at most 2k - 1, then reg( I (G)((s))) <= 2s + reg( I (G)) - 2, for every integer s <= k + 1. Finally, it is proven that reg( I (G)((s))) = 2s, for s. {2, 3, 4}, provided that the complementary graph (G) over bar is chordal.