Let P be a partial latin square of prime order p > 7 consisting of three cyclically generated transversals. Specifically, let P be a partial latin square of the form:for some distinct c, c ′ , c ′′ and some distinct s, s ′ , s ′′ . In this paper we show that any such P completes to a latin square which is diagonally cyclic.

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.

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There are well known necessary conditions for the existence of a generalized Bhaskar Rao design over a group $\mathbb{G}$, with block size $k=3$. We prove that they are sufficient for nilpotent groups $\mathbb{G}$ of even order, and in particular for $2$-groups. In addition, we prove that they are sufficient for semi-dihedral groups.

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