Combinatorics of the integral closure of edge ideals of strong quasi-n-partite graphs

Combinatorial properties of some ideals related to strong quasi-n-partites graphs are examined. We prove that the edge ideal of a strong quasi-n-partite graph is not integrally closed and we give an expression for its integral closure. Moreover, we are able to determine the structure of the ideals of vertex covers for the edge ideals associated to a strong quasi-n-partite graph.