Circular Coloring the Plane

In this paper we study the circular chromatic number of the unit distance graphR. Let r 2, a, b ∈ [0, r), and a b. We define the circular distance of a and b, denoted by δ(a, b) = δr(a, b), to be min{b−a, r+a−b}. One may identify the interval [0, r) with a circle C having circumference r, and then δ(a, b) will be the distance between a and b in C. It is easy to see that δ satisfies the triangle inequality. If a, b ∈ [0, r) (or equivalently a, b ∈ C), we define the circular interval from a to b, denoted [a, b], as follows (see Figure 1.1):