Magic labellings of infinite graphs over infinite groups

A total labelling of a graph over an abelian group is a bijection from the set of vertices and edges onto the set of group elements. A labelling can be used to define a weight for each edge and for each vertex of finite degree. A labelling is edge-magic if all the edges have the same weight and vertex-magic if all the vertices are finite degree and have the same weight. We exhibit magic labellings for various countable graphs. For countably infinite stars, countable galaxies and forests, and for an arbitrary abelian group, we determine when there is an edge-magic labelling and what group elements occur as constants. In the particular case where the group is the integers, we determine when there is an vertex-magic labelling, and which integers occur as constants. We also give explicit edge-magic labellings of various frieze graphs over the integers. AMS subject classification: 05C78