A 6-chromatic odd-distance graph in the plane

TL;DR

This paper constructs a specific type of graph in the plane where vertices are connected if their Euclidean distance is an odd integer, demonstrating that such graphs can require at least six colors for proper coloring.

Contribution

It introduces the first known example of a 6-chromatic odd-distance graph in the plane, advancing understanding of chromatic properties of geometric graphs.

Findings

Constructed a 6-chromatic odd-distance graph in the plane

Demonstrated the complexity of coloring such graphs

Provided new insights into geometric graph coloring problems

Abstract

Two vertices of an odd-distance graph are connected by an edge if and only if their Euclidean distance is an odd integer. We construct a 6-chromatic odd-distance graph in the plane.