Zigzag walk

A , also called a $1$-zigzag, is a polygon (usually skew) formed by following the edges of a polyhedron in a particular way.

Definition
A $k$-zigzag is a polygon formed by following the edges of a polyhedron in a particular way. Starting from a particular vertex and edge the polygon follows that edge to another vertex. It then follows along the $k$th rightmost edge to the next vertex, and then follows along the $k$th leftmost edge, repeating this pattern of left and right turns until the path loops back on itself.

A $k$-zigzag, that is a path alternating between the leftmost and rightmost possible edges, is also called a Petrie polygon or sometimes just a zigzag, and is integral in the construction of the Petrie dual.