Integral Closure of Powers of Edge Ideals of Weighted Oriented Graphs

Abstract

In this article, we study monomial ideals associated with a simple graph, namely edge ideals of weighted oriented graphs. Let D be a weighted oriented graph. Assuming that all the vertices of D have weights greater than 1, we completely characterize weighted oriented graphs D for which I(D) is integrally closed, and show that this is equivalent to I(D) being normal. We also give an equivalent condition for \(\overline{I(D)}=I(D)\) when the underlying simple graph of D is a complete graph. Finally, we give a necessary and sufficient condition when the edge ideal of a uniform whiskered graph is integrally closed.