Unit distance graphs
by Jan Kristian Haugland

Here are two unit distance graphs (UDGs) based on regular heptagons and heptagrams. I do not know if they are particularly useful, but I think they are quite appealing.
The one on the left* is 4-regular on 21 vertices, and is an induced subgraph of the one on the right* which is 6-regular on 63 vertices. Both have chromatic number 4.
(*Replace "the left" and "the right" with "top" and "bottom" if reading on a small screen.)

 


In contrast (symmetry-wise), here is the smallest UDG with chromatic number 4 for which the coordinates of the vertices can not all be

expressed as

with a, b, c, d integers of the same parity. (Implementing all the forbidden subgraphs sure was fun.)

A "poor man's Hadwiger-Nelson problem" is to ask for the minimal independence ratio of a UDG. The Moser spindle gives an
upper bound of 2/7, and it seems to be difficult to improve on this. I have not tested the ones with chromatic number 5 that have been
discovered in recent years. For a lower bound, take the intersection between a concentric circle of radius ever so slightly less than 1/2
and a regular hexagon of inradius r=0.482766155, and put copies in a triangular pattern with distance slightly more than 1+2r between
the centres, and flat sides facing each other. This gives an independent set of the whole plane of density 0.229364731629758...