Zonal graphs of small cycle rank

Andrew Bowling, Ping Zhang


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. 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.


planar graph, graph embedding, zonal labeling, zonal graph, cycle rank.

Full Text:


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


A. Bowling and P. Zhang, Absolutely and conditionally zonal graphs. Electron. J. Math. 4 (2022), 1–11.

A. Bowling and P. Zhang, Inner zonality in graphs Int. J. Comput. Math: Computer Systems Theory. 7 (2022), 192-206.

G. Chartrand, C. Egan, and P. Zhang, How to Label a Graph. Springer, New York (2019).

G. Chartrand, C. Egan, and P. Zhang, Zonal graphs revisited. Bull. Inst. Combin Appl. 99 (2023). To appear.

G. Chartrand and P. Zhang, Chromatic Graph Theory. Second Edition. Taylor & Francis/CRC Press, Boca Raton (2020).

K. Kuratowski, Sur le problème des courbes gauches en topologie. Fund. Math. 15 (1930), 271–283.


  • There are currently no refbacks.

ISSN: 2338-2287

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View EJGTA Stats