The current best known algorithm for recognizing map graphs (a generalization of planar graphs) runs in $n^{120}$. Thorup, Map graphs in polynomial time.Thorup, Map graphs in polynomial time.
Computing the equilibrium of the Arrow-Debreu market takes $O(n^6\log(nU))$ max-flow computations, where $U$ is the maximum utility. Duan, Mehlhorn, A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market.