Homomorphisms of Edge-Colored Graphs and Coxeter Groups

  • Authors:

  • N. Alon;T. H. Marshall

  • Affiliations:

  • Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. E-mail: noga@math.tau.ac.il.;School of Mathematical and Information Sciences, The University of Auckland, Private Bag 92019, Auckland, New Zealand. E-mail: t_marshall@math.auckland.ac.nz.

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 1998

Quantified Score

Hi-index 0.00

Visualization

Abstract

Let G_1=(V_1, E_1) 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 forwhich, for every pair of adjacent vertices u and vof G_1, ϕ(u) and ϕ(v) are adjacent inG_2 and the color of the edge ϕ(u)ϕ(v) isthe 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 everymember 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={m_1, m_2,..., m_k}, has a torsion-free subgroup ofindex not exceeding some bound, which depends only on the setM.