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 → 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, . . . , m_k], has a torsion-free subgroup of index not exceeding some bound, which depends only on the set M.