Uniform strong 2-cell imbeddings of bridgeless graphs Original Research Article

A strong 2-cell imbedding of a graph G is an open 2-cell imbedding of G in which the image of every edge of G separates distinct regions of the imbedding. An imbedding is uniform if every region is bounded by the same number of sides. A k-pattern is a uniform, strong 2-cell imbedding in which every region is bounded by exactly k sides. The existence of k-patterns for all values of k ⩾ 3 and all possible numbers of regions is illustrated for the sphere, the projective plane, the torus, and the Klein bottle. Characterizations of uniform imbeddings as k-patterns, based on the number of sides bounding each region (the index), are given. These characterizations are shown to be sharp.