## Edge-magic group labellings of countable graphs.(English)Zbl 1111.05085

Summary: We investigate the existence of edge-magic labellings of countably infinite graphs by abelian groups. We show that for a large class of abelian groups, including the integers $$\mathbb{Z}$$, 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 $$H + \mathcal I$$, where $$H$$ is some finite graph and $$\mathcal T$$ is a countable set of isolated vertices. Using power series of rational functions, we show that any edge-magic $$\mathbb Z$$-labelling of $$H + \mathcal I$$ has almost all vertex labels making up pairs of half-modulus classes. We also classify all possible edge-magic $$\mathbb Z$$-labellings of $$H + \mathcal I$$ under the assumption that the vertices of the finite graph are labelled consecutively.

### MSC:

 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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