Congestion in planar graphs with demands on faces.

link: http://arxiv.org/abs/1008.3653
Abstract

We give an algorithm to route a multicommodity flow in a planar graph $G$
with congestion $O(\log k)$, where $k$ is the maximum number of terminals on
the boundary of a face, when each demand edge lie on a face of $G$. We also
show that our specific method cannot achieve a substantially better congestion.