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{\bf Nicholas Cavenagh, Diana Combe and Adrian M. Nelson }
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{\bf Edge-Magic Group Labellings of Countable Graphs}
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We investigate the existence of edge-magic labellings of countably
infinite graphs by abelian groups. We show for that for a large class
of abelian groups, including the integers ${\Bbb Z}$, there is such a
labelling whenever the graph has an infinite set of disjoint edges. A
graph without an infinite set of disjoint edges must be some subgraph
of $H + {\cal I}$, where $H$ is some finite graph and ${\cal I}$ is a
countable set of isolated vertices. Using power series of rational
functions, we show that any edge-magic ${\Bbb Z}$-labelling of $H +
{\cal I}$ has almost all vertex labels making up pairs of half-modulus
classes. We also classify all possible edge-magic ${\Bbb
Z}$-labellings of $H + {\cal I}$ under the assumption that the
vertices of the finite graph are labelled consecutively.
\bye