Edge-magic group labellings of countable graphs
The electronic journal of combinatorics, Tome 13 (2006)
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.
@article{10_37236_1118,
author = {Nicholas Cavenagh and Diana Combe and Adrian M. Nelson},
title = {Edge-magic group labellings of countable graphs},
journal = {The electronic journal of combinatorics},
year = {2006},
volume = {13},
doi = {10.37236/1118},
zbl = {1111.05085},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1118/}
}
Nicholas Cavenagh; Diana Combe; Adrian M. Nelson. Edge-magic group labellings of countable graphs. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1118
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