  The Ising model S=1/2 and the S=1 model are studied by efficient Monte Carlo
schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The
algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering
protocol, are briefly described and compared with the simple Metropolis
algorithm. Accurate Monte Carlo data are produced at the exact critical
temperatures of the Ising model for these lattices. Their finite-size analysis
provide, with high accuracy, all critical exponents which, as expected, are the
same with the well known 2d Ising model exact values. A detailed finite-size
scaling analysis of our Monte Carlo data for the S=1 model on the same lattices
provides very clear evidence that this model obeys, also very well, the 2d
Ising model critical exponents. As a result, we find that recent Monte Carlo
simulations and attempts to define effective dimensionality for the S=1 model
on these lattices are misleading. Accurate estimates are obtained for the
critical amplitudes of the logarithmic expansions of the specific heat for both
models on the two Archimedean lattices.
