Symbolic power decompositions of disjoint cycle graphs

Abstract

In this paper, we study the connection between the odd cycles of a finite, simple graph [Formula: see text] and the symbolic powers of its edge ideal. When the odd cycles of [Formula: see text] are disjoint, we give a decomposition of the symbolic powers of the edge ideal based on the number and size of the odd cycles of [Formula: see text]. This shows that the symbolic powers of the edge ideal are completely dependent on the odd cycles of [Formula: see text].