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For values q < q$_{c}$ ≈ 0 . 35, a second-order transition takes place since in the lim$_{N}$ →∞ (2 / 3 − B$_{i,min}$ ) = 0, even at T$_{c}$ . However, for q > q$_{c}$ a first-order transition is observed, because one has (2 / 3 − B$_{i,min}$ ) ̸ = 0. For instance, for q = 0 . 4 we get (2 / 3 − B$_{min}$ ) = 0 . 273(3). 

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In Fig. 9 the fourth-order Binder cumulant is shown as a function of the K for several values of N and q = 0 . 1. Taking the largest lattices we have K$_{c}$ = 0 . 3109(6). To estimate U ∗ 4 we note that it varies little at K$_{c}$ so we have U ∗ 4 = 0 . 326(4). From the second-order cumu- lant (not shown) we similarly get K$_{c}$ = 0 . 31078(3) and U ∗ 2 = 0 . 538(2). For other values of q we have: for q = 0 . 2, K$_{c}$ = 0 . 3114(6) and U ∗ 4 = 0 . 326(5) and K$_{c}$ = 0 . 31164(5) and U ∗ 2 = 0 . 534(7); for q = 0 . 3, K$_{c}$ = 0 . 32154(8) and U ∗ 4 = 0 . 327(5), and K$_{c}$ = 0 . 32137(5) and U ∗ 2 = 0 . 556(4). One can see that the agreement of the critical temper- ature is quite good and U ∗ 4 is definitely different from the universal value U ∗ 4 ∼ 0 . 61 for the same model on the regular 2 D lattice and undirected Voronoi-Delaunay 

random lattice. 

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Fig. 10 displays the behavior of the magnetization fourth-order cumulant for q = 0 . 9 in a narrow range of K , where one can note a different behavior leading to a first-order phase transition. 

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In our analysis we also computed the modulus of the magnetization at the inflection point, T = T$_{c}$ for q = 0 . 1 , 0 . 2 , and 0 . 3 . The logarithm of these quantities as a function of the logarithm of L are presented in Figure 11. A linear fit of these data gives β/ν = 0 . 421(7) , 0 . 400(5) and 0 . 338(5) from the magnetization for q = 0 . 1 , 0 . 2, and 0 . 3, respectively. These results are totally different of β/ν = 0 . 125 obtained for a regular 2 D lattice. 

In Fig. 12 the logarithm plot of the susceptibility at T$_{c}$ as a function of the logarithm of L = N 1 / 2 is pre- sented. A linear fit of these data gives, at T = T$_{c}$ , γ/ν = 1 . 101(4) , 1 . 138(5) and 1 . 312(7). Similarly, from the maximum of the magnetic susceptibility (not shown) one gets γ/ν = 1 . 130(14) , 1 . 156(12) and 1 . 312(3) for 

