Absolutely and conditionally zonal graphs

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z3 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 Z3.A plane graph possessing such a labeling is a zonal graph.A planar graph G is zonal if there exists a zonal planar embedding of G.If every planar embedding of G is zonal, then G is absolutely zonal.A zonal planar graph G is conditionally zonal if some planar embedding of G is not zonal.It is shown that there is a class of absolutely zonal graphs possessing an arbitrarily large number of distinct zonal planar embeddings as well as a class of conditionally zonal graphs possessing an arbitrarily large number of distinct zonal planar embeddings with prescribed irregularity and regularity properties.

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