**Combinatorial Characterizations of the Saturation and the Associated Primes of the Fourth Power of Edge Ideals **

* Ha Thi Thu Hien, Ha Minh Lam *

**Abstract**

To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in Hien et al. (J. Algebra 439, 225–244, 2015) and Terai and Trung (J. Pure Appl. Algebra 218, 1117–1129, 2014). In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.