Abstrast: The unit distance graph is the infinite graph whose vertices are the points of the plane R^2, two vertices connected by an edge if their Euclidean distance is 1.
This graph was introduced by Ed Nelson in 1950. It was established in 1950 that the chromatic number $\chi(G)$ of this graph is between 4 and 7. In spite of many attempts, these bounds have not been improved.I introduced the odd distance graph in 1994.
Similarly, its vertices are the points in R^2, with two vertices connected by an edge if their Euclidean distance is an odd integer.In this talk I will describe what we know about this graph and related open problems.