A Pseudo-approximation for the Genus of Hamiltonian Graphs

The genus of a graph is a basic parameter in topological graph theory that has been the subject of extensive study. Perhaps surprisingly, despite its importance, the problem of approximating the genus of a graph is very poorly understood. Thomassen (1989) showed that computing the exact genus is NP-complete, and the best known upper bound for general graphs is an O(n)-approximation that follows by Euler’s characteristic. We give a polynomial-time pseudo-approximation algorithm for the orientable genus of Hamiltonian graphs. More specifically, on input a graph G of orientable genus g and a Hamiltonian path in G, our algorithm computes a drawing on a surface of either orientable or non-orientable genus O(g7). ACM Classification: F.2.2, G.2.2 AMS Classification: 68W25