The Odd-Distance Plane Graph
The vertices of the odd-distance graph are the points of the plane R(2). Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in R(2) is countably choosable, while such a graph in R(3) is not.
the unit-distance graph graph coloring list-chromatic number (choosability)