The performance analysis of dynamic routing algorithms in interconnection networks of parallel computers has thus far predominantly been done by simulation studies. A limitation of simulation studies is that they usually only hold for specific combinations of network, routing algorithm, and traffic pattern. In this paper, we derive saturation point results for the class of homogeneous traffic patterns and a large class of routing functions on meshes. We show that the best possible saturation point on a mesh is half the best possible saturation point on a torus. We also show that, if we restrict ourselves to homogeneous routing functions, the worst possible saturation point on a mesh is again half the best possible saturation point. Finally, we present a class of homogeneous routing functions, containing the well-known e-cube routing function, which are all optimal for all homogeneous traffic patterns.