As long as (p-2)(q-2) > 4 there exists a regular tiling of the hyperbolic plane H by regular hyperbolic p-gons that meet q per vertex. Suppose we have such a tiling with p = 5 and q >= 3. Then the union of all the edges of this tiling form a graph with countably many vertices and constant valence = q; label them using any bijection Z <—> {vertices}. —Dan _____ * Proof sketch: Think of the Poincaré disk model of H. Start with an arbitrarily small regular p-gon, whose interior angles will be arbitrarily close to Euclidean, i.e. (p-2)π/p, and slowly enlarge it. The angles will decrease toward 0 since geodesics meet the boundary of the unit disk at right angles. Stop when they are 2π/q. Since H is the same everywhere, the q tiles per vertex tiling can be continued indefinitely. ----- Allan Wechsler <acwacw@gmail.com> wrote: Find a connected graph of girth 5 (i.e. no loops of size 3 or 4) in which every integer is a vertex, every vertex has the same valence, and every vertex is part of a loop of size 5. -----