Associated primes and v-numbers of integral closures of powers of edge ideals

Abstract

Let $I=I(G)$ be the edge ideal of a finite simple graph $G$. We investigate the relationship between the associated primes and $v$-numbers of powers $I^t$ and their integral closures $\overline{I^t}$. For the third power, we establish the associated prime containment $\mathrm{Ass}(S/\overline{I^3})\subseteq \mathrm{Ass}(S/I^3)$, which yields $v(I^3)\le v(\overline{I^3})$. In contrast, for $t\ge 4$, we show that $\mathrm{Ass}(S/\overline{I^t})$ and $\mathrm{Ass}(S/I^t)$ are in general incomparable. Furthermore, for every fixed $t\ge 3$ and $r\ge 4$, we construct families of graphs for which the discrepancies $v(\overline{I^t})-v(I^t)$ and $v(I^r)-v(\overline{I^r})$ are arbitrarily large.