On the Existence of a Perfect Matching for 4-Regular Graphs Derived from Quadrilateral Meshes

In 1891, Peterson [6] proved that every 3-regular bridge-less graph has a perfect matching. It is well known that the dual of a triangular mesh on a compact manifold is a 3-regular graph. M. Gopi and D. Eppstein [4] use Peterson’s theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [3] elaborate on the creation of strips of quadrilaterals when a perfect matching exists. In general, not all 4-regular graphs have a perfect matching. An example planar, 4-regular graph without a perfect matching is given in this paper. However, in this paper, it is shown that the dual of a quadrilateral mesh on a 2-dimensional compact manifold with an even number of quadrilaterals (which is a 4-regular graph) always has a perfect matching.