On Maps with a Single Zigzag

If a graph GM is embedded into a closed surface S such that S\GM is a collection of disjoint open discs, then M = (GM , S) is called a map. A zigzag in a map M is a closed path which alternates choosing, at each star of a vertex, the leftmost and the rightmost possibilities for its next edge. If a map has a single zigzag we show that the cyclic ordering of the edges along it induces linear transformations, cP and cP∼ whose images and kernels are respectively the cycle and bond spaces (over GF (2)) of GM and GD, where D = (GD, S) is the dual map of M . We prove that Im(cP ◦ cP∼) is the intersection of the cycle spaces of GM and GD, and that the dimension of this subspace is connectivity of S. Finally, if M has also a single face, this face induces a linear transformation cD which is invertible: we show that c D = cP∼ .