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 qc≤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 qc=4. A lower bound on the first order critical temperature is additionally derived.
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