Orientable biembeddings of cyclic Steiner triple systems from current assignments on Mِbius ladder graphs

We give a characterization of a current assignment on the bipartite Mِbius ladder graph with 2 n + 1 rungs. Such an assignment yields an index one current graph with current group Z 12 n + 7 that generates an orientable face 2-colorable triangular embedding of the complete graph K 12 n + 7 or, equivalently, an orientable biembedding of two cyclic Steiner triple systems of order 12 n + 7 . We use our characterization to construct Skolem sequences that give rise to such current assignments. These produce many nonisomorphic orientable biembeddings of cyclic Steiner triple systems of order 12 n + 7 .