A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane

Abstract

With regard to the Hadwiger-Nelson problem, several 5-chromatic unit distance graphs in the Euclidean plane have been discovered in recent years. While most constructions rely heavily on the Moser spindle, a few recent examples completely avoid it, the smallest one consisting of 1441 vertices. In this note, we introduce an original geometric approach to constructing such graphs by utilizing the arcs of a 7-fold symmetric unit distance graph on 21 vertices, and obtain a Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices. While this is not a record small result, it arises from a straightforward, structured rule rather than a purely automated or brute-force search.