On the regularity of small symbolic powers of edge ideals of graphs

Abstract

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 $s\ge 1$, $$\mathrm{reg}(I(G{)}^{(s+1)})\le \max \{\mathrm{reg}(I(G))+2s,\mathrm{reg}(I(G{)}^{(s+1)}+I(G{)}^s)\}.$$As a consequence, we conclude that $\mathrm{reg}(I(G{)}^{(2)})\le \mathrm{reg}(I(G))+2$, and $\mathrm{reg}(I(G{)}^{(3)})\le \mathrm{reg}(I(G))+4$. Moreover, it is shown that if for some integer $k\ge 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then $\mathrm{reg}(I(G{)}^{(s)})\le 2s+\mathrm{reg}(I(G))-2$, for every integer $s\le k+1$. Finally, it is proven that $\mathrm{reg}(I(G{)}^{(s)})=2s$, for $s\in \{2,3,4\}$, provided that the complementary graph $\overline{G}$ is chordal.