On zonal and inner zonal labelings of plane graphs of maximum degree 3

Abstract

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 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 Z 3 . A plane graph possessing such a labeling is a zonal graph. If there is at most one exception, then the labeling is inner zonal and the graph is inner zonal. In 2019, Chartrand, Egan, and Zhang proved that showing the existence of zonal labelings in all cubic maps is equivalent to giving a proof of the Four Color Theorem. It is shown that every inner zonal cubic map is zonal, thereby establishing an improvement of the 2019 result. It is also shown that (i) while certain 2 -connected plane graphs of maximum degree 3 may not be zonal, they must be inner zonal and (ii) no connected cubic plane graph with bridges can be inner zonal.