On the penrose number of cubic diagrams

Abstract

A cubic diagram is a cubic graph G drawn in the plane, possibly with edge-crossings. The drawing defines a sign for each edge-3-coloring of G. The Penrose number of G is the sum of signs of its edge-3-colorings. For plane graphs it coincides with the number of edge-3-colorings. Given a cubic diagram G, we define a sign for every Eulerian orientation of its line-graph L( G) and prove that the Penrose number of G is equal to the sumof signs of the Eulerian orientations of L( G). This yields a new recursive scheme for the computation of the Penrose number. Another consequence is a simple formula which gives the number of vertex-4- colorings of a loopless plane triangulation in terms of the mappings from the vertex-set to {1,2,3} which take exactly two distinct values on each triangle.