Bilangan Kromatik Graceful pada Graf Terkait Lingkaran
DOI:
https://doi.org/10.30605/rj4gvz95Keywords:
Graceful Coloring, Graceful Chromatic Number, Sunflower Graph, Dutch Windmill GraphAbstract
Pewarnaan graceful pada graf adalah suatu bentuk pewarnaan pada titik yang menginduksi pewarnaan sisi berdasarkan nilai mutlak selisih antara dua titik yang bertetangga, di mana setiap titik dan sisi yang bertetangga memiliki warna yang berbeda. Bilangan kromatik graceful, dilambangkan dengan xg(G), adalah jumlah warna minimum yang memenuhi kondisi tersebut. Penelitian ini bertujuan untuk menentukan nilai eksak bilangan kromatik graceful pada dua variasi graf terkait lingkaran, yaitu graf bunga matahari (Sfn) dan graf kincir angin belanda (Dnm), yang memiliki struktur siklik dan konektivitas titik pusat yang khas. Pendekatan yang digunakan bersifat konstruktif-deduktif, yaitu dengan membangun fungsi pewarnaan titik secara eksplisit untuk setiap kasus parameter, kemudian membuktikan keoptimalannya melalui batas bawah yang diturunkan dari derajat maksimum graf. Hasil penelitian ini menunjukkan bahwa x_g(Sfn)=3n+1 untuk n>=3. Sementara itu, x_g(Dnm)=2m+1 untuk m>=2dan n>=3. Temuan ini memberikan pemahaman yang lebih jelas mengenai pola bilangan kromatik graceful pada graf terkait lingkaran dengan simpul pusat dominan serta memperluas karakterisasi teoretis untuk graf berbasis pusat dengan konektivitas radial.References
Annadhifi, M. I. N., Adawiyah, R., Dafik, D., & Suparta, I. N. (2024). Rainbow Vertex Connection Number Of Bull Graph, Net Graph, Triangular Ladder Graph, And Composition Graph (P_n [P_1 ]). BAREKENG: Jurnal Ilmu Matematika dan Terapan, 18(3), 1665–1672. https://doi.org/10.30598/barekengvol18iss3pp1665-1672
Budayana, I. N., Suparta, I. N., & Purnamayanti, N. P. H. (2018). Graceful labeling for open superstar of complete bipartite graphs S2(t•Km,n). Journal of Physics: Conference Series, 1040(1). https://doi.org/10.1088/1742-6596/1040/1/012020
Byers, A. D. (2018). ScholarWorks at WMU Graceful Colorings and Connection in Graphs Graceful Colorings and Connection in Graphs by.
Dwi, D., Lestari, A., Kristiana, A. I., Prihandini, R. M., Alfarisi, R., Setiawan, T. B., & Adawiyah, R. (2024). Bilangan Kromatik Graceful pada Keluarga Graf Sentripetal. J. DIFERENSIAL, 6(1), 57–64.
Firdausy, A. N., Dewi, P. K., Sudiarta, I. G. P., & Silalahi, R. Y. (2021). Math Unesa. Jurnal Ilmiah Matematika, 9(2), 437–446. https://media.neliti.com/media/publications/249234-model-infeksi-hiv-dengan-pengaruh-percob-b7e3cd43.pdf
Firman, F., Dafik, D., & Albirri, E. R. (2022). Rainbow Vertex Connection Number pada Keluarga Graf Roda. Cgant Journal of Mathematics and Applications, 3(1), 1–10. https://doi.org/10.25037/cgantjma.v3i1.71
Gallian, J. A. (2022). A Dynamic Survey of Graph Labeling. Electronic Journal of Combinatorics, 6(25), 4–623. https://doi.org/10.37236/11668
Javaid, M., Siddique, M. K., & Ali, U. (2021). Computational Journal of Combinatorial Mathematics Novel Connection Based Zagreb Indices of Dendrimer Nanostars. Comp. J. Combin. Math, 3(September 2021), 42–51. www.cojgco.com
Khoirunnisa, S., Dafik, Kristiana, A. I., & Alfarisi, R. (2021). Graceful Coloring of Wheel Graph Family. International Journal of Academic and Applied Research (IJAAR), 5(4), 68–78.
Kristiana, A. I., Anzori, Adawiyah, R., Slamin, & Albirri, E. R. (2023). Graceful Chromatic Number on Grid Graph Family. Jurnal Axioma: Jurnal Matematika dan Pembelajaran, 8(2), 144 – 155.
Suparta, I. N., & Ariawan, I. D. M. A. (2020). Expanding graceful trees. Electronic Journal of Graph Theory and Applications, 8(2), 217–232. https://doi.org/10.5614/ejgta.2020.8.2.2
Suparta, I. N., Dwi Novitarisa, A. A. A., & Suharta, G. P. (2022). Graceful Labeling of Some Join Graphs. 06(02), 129–144.
Suparta, I. N., Lin, Y., Hasni, R., & Budayana, I. N. (2025). On odd-graceful coloring of graphs. Communications in Combinatorics and Optimization, 10(2), 335–354. https://doi.org/10.22049/CCO.2023.28736.1692
Suparta, I. N., Venkathacalam, M., Gunadi, I. G. A., & Pratama, P. A. C. (2023). Graceful Chromatic Number of Some Cartesian Product Graphs. Ural Mathematical Journal, 9(2), 193–208. https://doi.org/10.15826/umj.2023.2.016
Utomo, R. B. (2024). Pewarnaan Graf Dengan Algoritma Welch-Powell Untuk Pengaturan Lampu Lalu Lintas Persimpangan Tugu Adipura Kota Tangerang. Buana Matematika: Jurnal Ilmiah Matematika dan Pendidikan Matematika, 14, 131–144.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Ni Kadek Ari Martadewi, Raphita Yanisari Silalahi, I Nengah Suparta
This work is licensed under a Creative Commons Attribution 4.0 International License.
In submitting the manuscript to the journal, the authors certify that:
- They are authorized by their co-authors to enter into these arrangements.
- The work described has not been formally published before, except in the form of an abstract or as part of a published lecture, review, thesis, or overlay journal.
- That it is not under consideration for publication elsewhere,
- That its publication has been approved by all the author(s) and by the responsible authorities – tacitly or explicitly – of the institutes where the work has been carried out.
- They secure the right to reproduce any material that has already been published or copyrighted elsewhere.
- They agree to the following license and copyright agreement.
License and Copyright Agreement
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
