  Using the tensor network approach, we investigate the monomer-dimer models on
a checkerboard lattice, in which there are interactions (with strength $\nu$)
between the parallel dimers on half of the plaquettes. For the fully packed
interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition
between the lowtemperature symmetry breaking and the high-temperature critical
phases; for the doped monomer-dimer casewith finite chemical potential $\mu$,
we also find an order-disorder phase transition which is of second order
instead. We use the boundary matrix product state approach to detect the KT and
second-order phase transitions and obtain the phase diagrams $\nu-T$ and
$\mu-T$ . Moreover, for the noninteracting monomer-dimer model (setting $\mu =
\nu = 0$), we get an extraordinarily accurate determination of the free energy
per site (negative of the monomer-dimer constant $h_2$) as $f=-0.662\, 798\,
972\, 833\, 746$ with the dimer density $n=0.638\, 123\, 109\, 228\, 547$, both
of 15 correct digits.
