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$\mathbb{Z}_k$-Zonal Labeling of Plane Graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 26 تیر 1405 اصل مقاله (2.88 M) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31469.2863 | ||
| نویسندگان | ||
| Fatemeh Eskandari1؛ Hassan Arianpoor* 2؛ Mohammad Habibi2 | ||
| 1Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran | ||
| 2Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| $\mathbb{Z}_k$-zonal labeling؛ inner $\mathbb{Z}_k$-zonal؛ planar graph؛ Halin graph؛ wheel graph | ||
| مراجع | ||
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[2] A. Bowling, Zonal and cozonal labelings using arbitrary abelian groups, Util. Math. 124 (2025), 83–100. https://doi.org/10.61091/um124-05 [3] A. Bowling and P. Zhang, Absolutely and conditionally zonal graphs, Electronic J. Math. 4 (2022), 1–11.
[4] A. Bowling and P. Zhang, Zonal graphs of small cycle rank, Electron. J. Graph Theory Appl. 11 (2023), no. 1, 1–14. https://dx.doi.org/10.5614/ejgta.2023.11.1.1 [5] F. Eskandari, H. Arianpoor, and M. Habibi, Zonal labeling of halin graphs, submitted.
[6] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. (2022), #DS6. https://doi.org/10.37236/11668
[7] P. Kovář, Magic labelings of graphs, Ph.D. thesis, VŠB–Technical University of Ostrava, 2004.
[8] G. Xin, X. Xu, C. Zhang, and Y. Zhong, On magic distinct labellings of simple graphs, J. Symbolic Comput. 119 (2023), 22–37. https://doi.org/10.1016/j.jsc.2023.02.005 | ||
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