Homomorphisms of edge-colored graphs and Coxeter groups

Let G(1) = (V-1, F-1) and G(2) = (V-2, E-2) be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mapping phi : V-1 \--> V-2 for which, for every pair of adjacent vertices u and v of G(1), phi(u) and phi(v) are adjacent in G(2) and the color of the edge phi(u)phi(upsilon) is the same as that of the edge u upsilon. We prove a number of results asserting the existence of a graph G,edge-colored from a set C,into which every member from a given class of graphs, also edge-colored from C, maps homomorphically. We apply one of these results to prove that every three-dimensional hyperbolic reflection group, having rotations of orders from the set M = {m(1), m(2),..., m(k)}, has a torsion-free subgroup of index not exceeding some bound, which depends only on the set M.