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

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## 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.

## Keywords

Edge ideal Power Associated prime Depth Dominating subgraph Odd cycle## Mathematics Subject Classification (2010)

13C05 13C15 13F55## Notes

### Acknowledgments

The authors would like to thank Prof. Ngo Viet Trung for his guidance.

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## Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015