Mixed product ideals and simplicial complexes

Let $I$ be a squarefree monomial ideal of the polynomial ring $K[x_1,\dots, x_n]$. We can consider $I$ as the Stanley Reisner ideal of a simplicial complex $\Delta_I$, on the set of vertices $x_1,\dots, x_n$, whose faces are defined by \[\Delta_I = \left\{\{x_{i_1}, \dots, x_{i_k}\}|i_1 < \dots < i_k, x_{i_1}\cdot \cdot \cdot x_{i_k} \notin I\left\} \qquad ).\] We are interested to special classes of squarefree monomial ideals of the polynomial ring $K[X,Y]$ in two sets of variables $X = \{x_1,\dots,x_n\}$ and $Y = \{y_1,\dots, y_m\}$, introduced first by Restuccia and Villarreal. More precisely, we study two classes of mixed product ideals, $I_rJ_r$ and $I_r +J_r$, where $I_r$ (resp. $J_r$) is the ideal of $K[X,Y]$ generated by all the squarefree monomials of degree $r$ in the variables $X$ (resp. $Y$) and the simplicial complexes associated to these ideals.