Overview
This note describes the construction of a 5-chromatic unit distance graph that is free of the Moser spindle. This graph exists in the Euclidean plane and comprises 2131 vertices. The development builds on previous work in the context of the Hadwiger-Nelson problem, specifically addressing constructions of 5-chromatic unit distance graphs.
Research Context
The Hadwiger-Nelson problem involves the determination of the chromatic number of the plane, often explored through the construction of unit distance graphs. In recent years, several 5-chromatic unit distance graphs have been identified within the Euclidean plane. Many of these constructions incorporate the Moser spindle. However, Voronov et al. previously presented examples of 5-chromatic unit distance graphs that do not contain the Moser spindle, though these examples involved 64513 vertices.
Approach
The construction detailed in this note aims to provide a 5-chromatic unit distance graph that is also Moser-spindle-free, while reducing the number of vertices compared to prior work. The methodology involves utilizing the arcs derived from a 7-fold symmetric unit distance graph. This foundational 7-fold symmetric graph is characterized by having 21 vertices.
Findings
The primary finding is the successful construction of a 5-chromatic unit distance graph on 2131 vertices. A key characteristic of this newly constructed graph is its complete absence of the Moser spindle. This represents an improvement over the earlier Moser-spindle-free examples by Voronov et al., which required 64513 vertices for their construction.