### 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 mapping φ : V_{1} → V_{2} for which, for every pair of adjacent vertices u and v of G_{1}, φ (u) and φ (v) are adjacent in G_{2} 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 = {m_{1}, m_{2}, . . . , m_{k}], has a torsion-free subgroup of index not exceeding some bound, which depends only on the set M.

Original language | English (US) |
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Pages (from-to) | 5-13 |

Number of pages | 9 |

Journal | Journal of Algebraic Combinatorics |

Volume | 8 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1998 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Keywords

- Coxeter group
- Graph
- Homomorphism
- Reflection group

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## Cite this

*Journal of Algebraic Combinatorics*,

*8*(1), 5-13. https://doi.org/10.1023/A:1008647514949