Comparison between regularity of small symbolic powers and ordinary powers of an edge ideal

Abstract

Let I=I(G) be the edge ideal of a simple graph G. We prove thatreg(I(s))=reg(Is) for s=2,3, where I(s) is the s-th symbolic power of I. As a consequence, we prove the following boundsregIs≤regI+2s−2, for s=2,3,regI(s)≤regI+2s−2, for s=2,3,4.