On Maps with a Single Zigzag

Abstract

If a graph $G_M$ is embedded into a closed surface $S$ such that $S \backslash G_M$ is a collection of disjoint open discs, then $M=(G_M,S)$ is called a {\em map}. A {\em 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, $c_P$ and $c_{P^\sim}$ whose images and kernels are respectively the cycle and bond spaces (over GF(2)) of $G_M$ and $G_D$, where $D=(G_D,S)$ is the dual map of $M$. We prove that $Im(c_P \circ c_{P^\sim})$ 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 $c_D^{-1} = c_{P^\sim}$.