Circular Coloring the Plane
Matt DeVos, Javad B. Ebrahimi, M. Ghebleh, Luis Goddyn, Bojan Mohar, Reza Naserasr · SIAM Journal on Discrete Mathematics · 2007
The unit distance graph $\mathcal{R}$ is the graph with vertex set $\mathbb{R}^2$ in which two vertices (points in the plane) are adjacent if and only if they are at Euclidean distance 1. We prove that the circular chromatic number of $\mathcal{R}$ is at least 4, thus improving the known lower bound of $32/9$ obtained from the fractional chromatic number of $\mathcal{R}$.