TY - JOUR

T1 - Homomorphisms of Edge-Colored Graphs and Coxeter Groups

AU - Alon, N.

AU - Marshall, T. H.

N1 - Funding Information:
⁄Research supported in part by a USA Israeli BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.

PY - 1998

Y1 - 1998

N2 - 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, . . . , mk], has a torsion-free subgroup of index not exceeding some bound, which depends only on the set M.

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

KW - Coxeter group

KW - Graph

KW - Homomorphism

KW - Reflection group

UR - http://www.scopus.com/inward/record.url?scp=0039527992&partnerID=8YFLogxK

U2 - 10.1023/A:1008647514949

DO - 10.1023/A:1008647514949

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AN - SCOPUS:0039527992

SN - 0925-9899

VL - 8

SP - 5

EP - 13

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 1

ER -