Bilangan Terhubung Pelangi pada Graf Tengah (M(G)) dari Graf Ulat (C_(m,2))
DOI:
https://doi.org/10.55657/rmns.v4i1.204Keywords:
Rainbow Connection Number, Middle Graph, Caterpillar GraphAbstract
Edge coloring of a graph is considered rainbow connected if the graph is connected and a rainbow path exists for every pair of points. The rainbow connection number of a graph, denoted as , represents the smallest number of colors required to make the graph is rainbow connected. This study examines the rainbow connection number of the middle graph of a caterpillar graph. The middle graph is a modified result of a graph , denoted as . It is described as a graph constructed from the intersection of a set of points and edges. The set of points in the middle graph consists of the combination of points and edges of the graph . Two points are considered adjacent if only they are connected in , or if one point corresponds to a point and the other corresponds to an edge adjacent to it. A caterpillar graph denoted by is a tree that will be a path if all the leaf points are deleted. The results of this research show the rainbow-connected number theorem for the middle graph of the caterpillar graph for .
Downloads
References
[1] M. Afifuddin and I. K. Budayasa, “Bilangan Keterhubungan Titik Pelangi Kuat Pada Graf,” MATHunesa J. Ilm. Mat., vol. 10, no. 1, pp. 33–40, 2022, doi: 10.26740/mathunesa.v10n1.p33-40.
[2] Afriantini, Helmi, and F. Fran, “Pewarnaan Simpul, Sisi, Wilayah Pada Graf Dan Penerapannya,” Bimaster Ilmiah. Stat. dan Ter., vol. 8, no. 4, pp. 773–782, 2019, doi: 10.26418/bbimst.v8i4.36037.
[3] D. J. Panjaitan and R. Aprilia, Teori Graph. Medan: LPPM UMNAW, 2022.
[4] M. S. Prof. Dr. Hasmawati, Pengantar dan jenis-jenis graf. Unhas Press, 2020.
[5] A. Darmawahyuni and Narwen, “Bilangan Kromatik Lokasi Dari Graf Ulat,” J. Mat. FMIPA UNAND, vol. 7, no. 1, p. 43, 2016.
[6] S. D. Sancoko, “Pelabelan Antiajaib Jarak Pada Beberapa Kelas Graf Terkait Graf Helm,” J. Ris. dan Apl. Mat., vol. 4, no. 2, p. 93, 2020, doi: 10.26740/jram.v4n2.p93-102.
[7] N. Y. Sari, E. Noviani, and F. Fran, “Pelabelan Fibonacci Prima Ke-k Pada Graf H dan Graf Ulat H_n,” KUBIK J. Publ. Ilm. Mat., vol. 8, no. 2, pp. 89–98, 2023, doi: 10.15575/kubik.v8i2.29290.
[8] G. Chartrand et al., “Rainbow connection in graphs,” J. Comb. Math. Comb. Comput., 2008.
[9] S. A. Farihati, A. N. M. Salman, and P. E. Putri, “Rainbow connection numbers of some classes of s-overlapping r-uniform hypertrees with size t,” AIMS Math., vol. 9, no. 7, pp. 18824–18840, 2024, doi: 10.3934/math.2024916.
[10] D. Fitriani, A. N. M. Salman, and Z. Y. Awanis, “Rainbow connection number of comb product of graphs,” Electron. J. Graph Theory Appl., vol. 10, no. 2, pp. 461–473, 2022, doi: 10.5614/ejgta.2022.10.2.9.
[11] Y. Joko, Helmi, and F. Fran, “Bilangan Terhubung Pelangi Pada Graf Planter Dan Graf Gurita,” Bimaster Bul. Ilm. Mat. Stat. dan Ter., vol. 8, no. 1, pp. 29–34, 2019, doi: 10.26418/bbimst.v8i1.30508.
[12] K. Q. Fredlina, A. N. M. Salman, I. G. P. K. Julihara, K. T. Werthi, and N. L. P. N. S. P. Astawa, “Rainbow Coloring of Three New Graph Classes,” J. Phys. Conf. Ser., vol. 1783, no. 1, 2021, doi: 10.1088/1742-6596/1783/1/012033.
[13] C. A. P. Noor, L. Yahya, S. K. Nasib, and N. I. Yahya, “BILANGAN TERHUBUNG PELANGI PADA GRAF SALJU,” J. Fundam. Math. Appl., vol. 4, no. 1, pp. 29–44, 2021, doi: 10.14710/jfma.v4i1.9035.
[14] T. Hamada and I. Yoshimura, “Traversability and Connectivity of the Middle Graph of A Graph,” vol. 1974, no. June, 1974.
[15] A. Tamilselvi and M. Ponni, “Middle Graph of Semiring Valued Graphs,” Turkish World Math. Soc. J. Appl. Eng. Math., vol. 13, no. 2, pp. 765–772, 2023.
[16] K. S. Rao and R. Murali, “Rainbow Connection Number of Sunlet Graph and its Line , Middle and Total Graph,” Int. J. Math. its Appl., vol. 3, no. 4, pp. 105–113, 2015.
[17] Y. Zhao, S. Li, and S. Liu, “(Strong) Rainbow Connection Number of Line, Middle and Total Graph of Sunlet Graph,” Proc. 2018 IEEE Int. Conf. Prog. Informatics Comput. PIC 2018, pp. 175–179, 2018, doi: 10.1109/PIC.2018.8706299.
[18] D. Rahmawati, F. Fran, Yudhi, and D. Krisantoni, “Total Rainbow Connection Number of n-Centipede Graph and Its Line, Square, and Middle Graph,” AIP Conf. Proc., vol. 040007, no. September, 2020.
[19] T. Aruna and S. Jeyakokila, “Some results on Rainbow Connection Number of Graphs,” Adv. Appl. Math. Sci., vol. 22, no. 8, pp. 1743–1753, 2023.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Fuji Fauzia Kiayi, Sumarno Ismail, Nisky Imansyah Yahya, Lailany Yahya, Salmun K. Nasib

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