### Zonal graphs of small cycle rank

#### 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. There is a connection between zonal labelings of connected bridgeless cubic plane graphs and the Four Color Theorem. Zonal labelings of cycles play a role in this connection. The cycle rank of a connected graph of order *n* and size *m* is *m* − *n* + 1. Thus, cycles have cycle rank 1. All zonal connected graphs of cycle rank at most 2 are determined.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2023.11.1.1

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