On the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph

Let $n\ge m$ be two positive integers, $S_{n,m}=K[x_1,\dots ,x_n,y_1,\dots ,y_m]$ and $I_{n,m}=(x_i{y}_j:1\le i\le n,1\le j\le m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\mathrm{hdepth}(S_{n,m}\mathrm{/}{I}_{n,m})$. We prove that $h(n,m)\ge \lceil \frac{n}{2}\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\lceil \frac{n}{2}\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m^{\mathrm{\prime }})$.