On Maps with a Single Zigzag

Abstract

If a graph G_M is embedded into a closed surface S such that S\G_M is a collection of disjoint open discs, then M = 3D(G_M, 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 G_M and G_D, where D= 3D(G_D, S) is the dual map of M. We prove that Im(cp o cp∼) is the intersection of the cycle spaces of G_M and G_D, 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 c_D which is invertible: we show that C^-1_D = 3Dcp∼.