  A combinatorial approach is used to study the critical behavior of a
$q$-state Potts model with a round-the-face interaction. Using this approach it
is shown that the model exhibits a first order transition for $q>3$. A second
order transition is numerically detected for $q=2$. Based on these findings, it
is deduced that for some two-dimensional ferromagnetic Potts models with
completely local interaction, there is a changeover in the transition order at
a critical integer $q_c\leq 3$. This stands in contrast to the standard
two-spin interaction Potts model where the maximal integer value for which the
transition is continuous is $q_c=4$. A lower bound on the first order critical
temperature is additionally derived.
