Edge-Magic Group Labellings of Countable Graphs

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.