Randomized Approximation Schemes for the Tutte Polynomial and Random Clustering in Subdense and Superdense Graphs

Extending the work of Alon, Frieze abnd Welsh, we show that there are randomized polynomial time approximation schemes for computing the Tutte polynomial in subdense graphs with an minimal node degree of $\Omega\left ( \frac{n}{\sqrt{\log n}}\right )$ . The same holds for the partition function $Z$ in the random cluster model with uniform edge probabilities and for the associated distribution $\lambda (A),\: A \subseteq E$ whenever the underlying graph $G=(V,E)$ is $c\cdot\frac{n}{\sqrt{\log (n)}}$-subdense. In the superdense case with node degrees $n-o(n)$, we show that the Tutte polynomial $T_G(x,y)$ is asymptotically equal to $Q=(x-1)(y-1)$. Moreover, we briefly discuss the problem of approximating $Z$ in the case of $(\alpha, \beta )$-power law graphs.