These are a simple animated visualizations I created while studying the problem. They contain a sequence of 2s frames starting at 130 vertices and incrementing 62. The red numbered text are the first few starting vertex numbers showing the hexagonal structure of the construction. On the left is a list of sorted and color coded vertices by the number of edges on a vertex for that frame’s shown vertices (it also lists in the right column the number of vertices with that number of edges & color). The solution data was obtained from: https://github.com/marijnheule/CNP-SAT

Here is a link to a Mathematica Notebook (50Mb) that has interactive ListAnimated graphs with vertex data (vertex number, symbolic and numeric coordinates, number of edges on the vertex in the frame) in Tooltip on mouseover.