Equality of Ordinary and Symbolic Powers of Some Classes of Monomial Ideals

In this article, our aim is to extend the class of monomial ideals for which symbolic and ordinary powers coincide. This property has been characterized for the class of edge ideals of simple graphs, and in this article, we study a completely new class of monomial ideals associated to simple graphs, namely the class of generalized edge ideals. We give a complete description of the primary components associated to the minimal associated primes of these ideals. Using this description, and assuming some conditions on the relative weights, we completely characterize the equality of ordinary and symbolic powers of generalized edge ideals. After that, we also characterize generalized edge ideals of the 3-cycle for which this equality holds.