$\mathbb{Z}_k$-Zonal Labeling of Plane Graphs

Articles in Press

Document Type : Original paper

Authors

Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran

Abstract

‎Zonal labeling in plane graphs plays a key role in the structural analysis of graph faces and is closely related to the Four Color Theorem‎. ‎In this paper‎, ‎we extend the classical zonal labeling framework from the modular group $\mathbb{Z}_3$‎​ ‎to $\mathbb{Z}_k​$ for all $k \geq 2$‎. ‎We show that this extension preserves the essential properties of zonality and provides increased flexibility for studying a broader class of plane graphs‎. ‎We establish the necessary and sufficient conditions for $\mathbb{Z}_k$​-zonal labeling and examine the corresponding inner zonal structures‎. ‎We also characterize the implementation of inner zonal labeling within Halin graphs‎. ‎This extended framework offers clear insights and practical methods for a deeper understanding of zonal characteristics in planar graph theory‎.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 17 July 2026

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