There is a class of proximity graphs parameterized by a real number β that are called β-skeletons and which are geometrical graphs whose vertex set V is a discrete set of points in the plane and two points p and q are adjacent if certain region luneβ(p, q) defined by them does not contain any other vertex. Special cases of β-skeletons are the Gabriel Graph (β = 1) and the Relative Neighborhood Graph (β = 2). Cyclic tilings are tilings of the plane whose tiles are convex cyclic polygons that contain the center of the circle in which they are inscribed. Up to our knowledge, this class of tilings has not been studied before. They generalize regular, Archimedean, and k-uniform tilings. We give sufficient conditions for the graph of a cyclic tiling to be a β-skeleton of its vertices. The result is particularly nice for the case of the Gabriel graph. The study of the relationship between tilings arises while studying street networks. This relationship is illustrated briefly.