Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus n with k punctures for any n and k.
To appear in Israel Journal of Mathematics, 2021. Slightly longer arXiv version.
Here are some examples:
1. (a.) On the 3-punctured sphere. Vertices=3, Tracks=1.
This is the Hass-Scott
(b.) On the punctured torus. Vertices=3, Tracks=3.
2. On the punctured torus. (a.) Vertices=7, Tracks=1. (b.) Vertices=8, Tracks=2.
3. On the 2-holed torus. Vertices=9, Tracks=1.
tony AT math.stonybrook.edu
September 28, 2021