Plane graphs with Eulerian Petrie walks Original Research Article

A Petrie walk in a plane graph G is obtained by walking on edges of G, alternatingly selecting as next edge the left edge and the right edge of the current edge in the rotation of edges around the common vertex. We give a characterization of 4-valent plane graphs with Eulerian Petrie walks, which gives rise to a simple algorithm for constructing such graphs. This algorithm is then used to answer the question whether or not there exist 4-valent plane graphs with Eulerian Petrie walks and faces of prescribed sizes.