Edge-magic group labellings of countable graphs (Q870009)

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.