| Let $G_1=(V_1, E_1)$ and $G_2 = (V_2, E_2)$ be two edge-coloured graphs (without multiple edges or loops). A {em homomorphism} is a mapping $phi: V_1 longmapsto V_2$ for which, for every pair of adjacent vertices $u$ and $v$ of $G_1, phi(u)$ and $phi(v)$ are adjacent in $G_2$ and the colour of the edge $phi(u)phi(v)$ is the same as that of the edge $uv$. We prove a number of results asserting the existence of a graph $G$, edge-coloured from a set $C$, into which every member from a given class of graphs, also edge-coloured from $C$, maps homomorphically. We apply one of these results to prove that every 3-dimensional hyperbolic reflection group, having rotations of orders from the set $M={m_1, m_2, ldots m_k }$, has a torsion-free subgroup of index not exceeding some bound, which depends only on the set $M$. |
Keywords
Math Review Classification
Primary 05C15, 05C25, 20F55, 51F15
Last Updated
21/3/97
Length
9 pages
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