Homomorphisms of Edge-Colored Graphs and Coxeter Groups

AbstractLet $$G_1 = (V_1 ,E_1 ){\text{ and }}G_2 = (V_2 ,E_2 )$$ be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mappingϕ : $$V_1 \mapsto V_2 $$ 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 graphG , 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 setM ={m1, m2,..., mk}, has a torsion-free subgroup of index not exceeding some bound, which depends only on the setM .