It turns out that there is even more to say on the communes of Lichtenstein.
First of all, there is a 5-clique in the communal graph, as Brian Hayes pointed out. But there are two different excluded subgraphs for planarity – so if we aren’t looking specifically for the chromatic number, but rather how this graph fails to be a “normal” land map, we might want to see whether it realizes BOTH.
It turns out that it does.
The following are two highlighted versions of the Liechtenstein communal graph.
The embedded K5 with edges in blue.
The embedded K33 with blue and red vertices.
(105 words, 2 images)
Following the featuring of the internal political structure of Lichtenstein on the Strange Maps blog, Brian Hayes asks for the chromatic number of Lichtenstein.
Rahul pointed out that I made errors in transferring the map to a graph. Specifically, I missed the borders Schellenberg-Eschen and Vaduz-Triesen. The post below changes accordingly.
Warning: This post DOES contain spoilers to Brian’s question. If you do want to investigate it yourself, you’ll need to stop reading now. Apologies to those on my planet feeds.
As a first step, we need to build a graph out of it. I labeled each region in turn with the exclaves numbered higher than the “main” region of each organizational unit. And then I build a .dot file to capture them all:
This is a preview of
On the chromatic number of Lichtenstein
. Read the full post (580 words, 4 images)