 
Summary: Homomorphisms of Edgecoloured Graphs and Coxeter Groups
N. Alon
T. H. Marshall
Abstract
Let G1 = (V1, E1) and G2 = (V2, E2) be two edgecoloured graphs (without multiple edges or
loops). A homomorphism is a mapping : V1  V2 for which, for every pair of adjacent vertices
u and v of G1, (u) and (v) are adjacent in G2 and the colour of the edge (u)(v) is the same
as that of the edge uv.
We prove a number of results asserting the existence of a graph G, edgecoloured from a
set C, into which every member from a given class of graphs, also edgecoloured from C, maps
homomorphically.
We apply one of these results to prove that every hyperbolic reflection group, having rotations
of orders from the set M = {m1, m2, . . . mk}, has a torsionfree subgroup of index not exceeding
some bound, which depends only on the set M.
1 Introduction
Let G1 = (V1, E1) and G2 = (V2, E2) be two edgecoloured graphs (without multiple edges or loops).
We define a mapping : V1  V2 to be a homomorphism if, for every pair of adjacent vertices u
and v of G1, (u) and (v) are adjacent in G2 and the colour of the edge (u)(v) is the same as
that of the edge uv. In Section 2 we prove a number of results asserting the existence of a graph G,
edgecoloured from a set C, into which every graph from a given class of graphs, also edgecoloured
