Homomorphisms of Edge-Colored Graphs and Coxeter Groups

Let G1 = (V1, E1) and G2 = (V2, E2) be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mapping φ : V1 7→ V2 for which, for every pair of adjacent vertices u and v of G1, φ(u) and φ(v) are adjacent in G2 and the color of the edge φ(u)φ(v) is the same as that of the edge uv. 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 = {m1,m2, . . . ,mk}, has a torsion-free subgroup of index not exceeding some bound, which depends only on the set M .