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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Ping Zhang
Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA
Electrical & Electronics Engineering

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Zonal graphs of small cycle rank Andrew Bowling; Ping Zhang
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 11, No 1 (2023): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2023.11.1.1

Abstract

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.
Co-Authors Andrew Bowling