Inner zonality in graphs

A zonal labelling of a plane graph G is an assignment of the two nonzero elements of the ring of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of . A plane graph possessing such a labelling is a zonal graph. A cubic map is a connected 3-regular bridgeless plane graph. It is known that if an independent proof could be given that every cubic map is zonal, then the Four Color Theorem would follow as a corollary. As a step in this direction, it is shown that certain subgraphs of cubic maps are nearly zonal.