Sketch of 10-vertex unit distance graph, 2013
In 2013, I wrote a program to search for unit distance graphs with as many edges as possible, for the number of vertices, and that take 4 colors to color the vertices so that vertices connected by an edge are colored differently.
In a unit distance graph, or rather its embedding in the plane, all edges between vertices have length 1, a.k.a. unit length.
Some sci.math readers including myself came to the conclusion that there were 19 edges of length exactly 1, that it was geometrically feasible with the 10 vertices. James Waldby helped to understand the geometry better in his post in the thread. The entire thread, “another unit distance graph (10 vertices)”, is archived at the Math Forum at the URL:
In my original post, I mention graphing the figure on graph paper. Below, a scan of that figure is reproduced.