  Let $(R,\mm)$ be a Noetherian local ring and $M$ a finitely generated
$R$-module. We say $M$ has maximal depth if there is an associated prime $\pp$
of $M$ such that $\depth M=\dim R/\pp$. In this paper we study squarefree
monomial ideals which have maximal depth. Edge ideals of cycle graphs,
transversal polymatroidal ideals and high powers of connected bipartite graph
with this property are classified.
