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Super (a,d)-P_2⨀P_m-Antimagic Total Labeling of Corona Product of Two Paths | Yatin | JTAM (Jurnal Teori dan Aplikasi Matematika)

Super (a,d)-P_2⨀P_m-Antimagic Total Labeling of Corona Product of Two Paths

Bela Zainun Yatin, Zata Yumni Awanis, I Gede Adhitya Wisnu Wardhana

Abstract


Graph labeling involves mapping the elements of a graph (edges and vertices) to a set of positive integers. The concept of an anti-magic super outer labeling (a,d)-H pertains to assigning labels to the vertices and edges of a graph using natural numbers {1,2,3,...,p+q}. The weights of the outer labels H form an arithmetic sequence {a,a+d,a+2d,...,a+(k-1)d}, where 'a' represents the first term, 'd' is the common difference, and 'k' denotes the total number of outer labels, with the smallest label assigned to a vertex. This study investigates the super (a,d)-P_2⨀P_m-antimagic total labeling of the corona product P_n⨀P_m, where n and m are both greater than or equal to 3. We define the labeling functions for vertices and edges based on the partitioning of labels into three subsets. Using k-balanced and (k,δ)-anti balanced multisets, we demonstrate that for m being odd, P_n⨀P_m is super (9m^2 n+4mn+m-n+3,1)-P_2 ⨀▒P_(m ) -antimagic, and for m being even, P_n⨀P_m is super (9m^2 n+4mn+m-2n+5,3)-P_2 ⨀▒P_(m ) -antimagic. The labeling scheme is illustrated through examples. For the case when m is odd, an antimagic total labeling of P_3 ⨀▒P_3    forms a super (282,1)- P_2 ⨀▒P_(3 )  -antimagic labeling. In the case of even m, an antimagic total labeling of P_3 ⨀▒P_(4 ) results in a super (483,3)- P_2 ⨀▒P_(4 )  -antimagic labeling. Both of these examples provide insights into the antimagic properties of corona products.

Keywords


(a,d)-H-antimagic total labeling; Corona product; H-covering; Path; Super (a,d)- H-antimagic total labeling.

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References


Agustin, I. H., Prihandini, R. M., & Dafik. (2019). P_2▹H-super antimagic total labeling of comb product of graphs. AKCE International Journal of Graphs and Combinatorics, 16(2), 163–171. https://doi.org/10.1016/j.akcej.2018.01.008

Anjaneyulu, G. S. G. N., Vijayabarathi, A., & Liu, L. (2015). Super edge-magic sequence of maximal outer planer graph and its characteristics. Cogent Mathematics, 2(1), 1123340. https://doi.org/10.1080/23311835.2015.1123340

Bača, M., Miller, M., Ryan, J., & Semaničová-Feňovčíková, A. (2019). Magic and Antimagic Graphs (1st ed., Vol. 60). Springer International Publishing. https://doi.org/10.1007/978-3-030-24582-5

Chang, F., Chen, H., & et al. (2019). Shifted-Antimagic Labelings for Graphs. Graphs and Combinatorics, 37(2021), 1065–1082, https://doi.org/10.1007/s00373-021-02305-w.

Diestel, R. (2017). Graph Theory (5th ed., Vol. 173). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-53622-3

Enomoto, H., Llado, A. S., Nakamigawa, T., & Ringel, G. (1998). Super Edge-Magic Graphs. SUT Journal of Mathematics, 34(2). https://doi.org/10.55937/sut/991985322

Gallian, J. A. (2022). A Dynamic Survey of Graph Labeling. In the electronic journal of combinatorics, https://www.combinatorics.org/files/Surveys/ds6/ds6v25-2022.pdf.

Gutiérrez, A., & Lladó, A. (2005). Magic coverings. Ournal of Combinatorial Mathematics and Combinatorial Computing, 55, 43, https://www.researchgate.net/publication/266257368_Magic_coverings.

Hasni, R., Tarawneh, I., & Husin, M. Na. (2022). A Remark On The Edge Irregularity Strength Of Corona Product Of Two Paths. Journal of Mathematical Sciences and Informatics, 2(1), 51–58. https://doi.org/10.46754/jmsi.2022.06.005

Hendy, H., Sugeng, K. A., & Salman, A. N. M. (2018). An H -super magic decompositions of the lexicographic product of graphs. AIP Conference Proceedings, 2023. https://doi.org/10.1063/1.5064190

Hendy, H., Sugeng, K. A., Salman, A. N. M., & Ayunda, N. (2018). Another H-super magic decompositions of the lexicographic product of graphs. Indonesian Journal of Combinatorics, 2(2), 72. https://doi.org/10.19184/ijc.2018.2.2.2

Hussain, M., Baskoro, E. T., Ali, K., & Hussain, M. (2012). On super antimagic total labeling of Harary graph. https://www.researchgate.net/publication/228458682

Inayah, N., Salman, A. N. M., & Simanjuntak, R. (2009). On (a, d)-H-antimagic coverings of graphs, Paper_Ina_JMCC-libre.pdf (d1wqtxts1xzle7.cloudfront.net).

Inayah, N., Simanjuntak, R., Salman, A. N. M., & Syuhada, K. I. A. (2013). Super (a, d)-H-antimagic total labelings for shackles of a connected graph H. In Australasian Journal Of Combinatorics (Vol. 57), 127-138, L175.dvi (uq.edu.au).

Indunil, minusha, & Perera.A.A.I. (2022). Graceful Labeling of Chain Graphs with Pendants. Nternational Conference On Business Innovations 2022, 595–598, https://nspace.nsbm.ac.lk/handle/123456789/213.

Kotzig, A., & Rosa, A. (1970). Magic Valuations of Finite Graphs. Canadian Mathematical Bulletin, 13(4), 451–461. https://doi.org/10.4153/cmb-1970-084-1

Liu, J. B., Afzal, H. U., & Javaid, M. (2020). Computing Edge Weights of Magic Labeling on Rooted Products of Graphs. Mathematical Problems in Engineering, 2020. https://doi.org/10.1155/2020/2160104

Llado A, & Moragas J. (2007). Cycle-Magic Graphs. DiscreteMath, 307, 2925–2933, https://doi.org/10.1016/j.disc.2007.03.007.

Martin Bača, Martin Bača, Rinovia Simanjuntak, & Rinovia Simanjuntak. (2006). Super edge-antimagic labelings of the generalized Petersen graph P(n,(n-1)/2). Utilitas Mathematica, 70, 119–127, https://www.researchgate.net/publication/262723084_Super_edge-antimagic_labelings_of_the_generalized_Petersen_graph_Pnn-12.

Maryati, T. K., Salman, A. N. M., & Baskoro, E. T. (2013). Supermagic coverings of the disjoint union of graphs and amalgamations. Discrete Mathematics, 313(4), 397–405. https://doi.org/10.1016/j.disc.2012.11.005

Ngurah, A. A. G., Salman, A. N. M., & Susilowati, L. (2010). H-supermagic labelings of graphs. Discrete Mathematics, 310(8), 1293–1300. https://doi.org/10.1016/j.disc.2009.12.011

Nurvazly, D. E., Chasanah, S. L., & Wiranto, A. R. (2022). Variations of graceful labelling of subgraph of millipede graph. AIP Conference Proceedings, 050018. https://doi.org/10.1063/5.0105447

Palanivelu, C., & Neela, N. (2019). Super (a, d)-Edge Antimagic Total Labeling of Union of Stars. In International Journal of Applied Engineering Research (Vol. 14, Issue 9). 2089-2092 http://www.ripublication.com

Permata Sari, A., Sri Martini, T., & Yugi Kurniawan, V. (2019). Super (a, d)-H-antimagic total labeling of edge corona product on cycle with path graph and cycle with cycle graph. Journal of Physics: Conference Series, 1306(1), 1-6, https://doi.org/10.1088/1742-6596/1306/1/012006

Prihandini, R. M., & Adawiyah, R. (2022). On Super (a,d)-Edge Antimagic Total Labeling of Some Generalized Shackle of Fan Graph. In International Journal of Academic and Applied Research (Vol. 6, issue. 4), 28-32, www.ijeais.org/ijaar.

Prihandini, R. M., & Adawiyah, R. (2023). On Super (3n+5,2)- Edge Antimagic Total Labeling And It’s Application To Construct Hill Chiper Algorithm. BAREKENG: Jurnal Ilmu Matematika Dan Terapan, 17(1), 0029–0036. https://doi.org/10.30598/barekengvol17iss1pp0029-0036

Raheem, A., Baig, A. Q., & Javaid, M. (2014). On Super (a, d)-Edge-Antimagic Total Labeling of a Class of Trees. In International J.Math. Combin (Vol. 4), 1-6, https://mathcombin.com/upload/file/20150127/1422320940239094100.pdf#page=52.

Rajeswari, R., Aruna Sakthi, K., & Meenakumari, N. (2021). Product-Set Labeling of Some Zero-Divisor Graphs and its Line Graph. Journal of Physics: Conference Series, 1947(1), 1-7 https://doi.org/10.1088/1742-6596/1947/1/012025

Sandariria, H., Roswitha, M., & Kusmayadi, T. A. (2017). H-Supermagic Labeling on Coronation of Some Classes of Graphs with a Path. Journal of Physics: Conference Series, 855(1), 1-6. https://doi.org/10.1088/1742-6596/855/1/012042

Simanihuruk, M., Kusmayadi, T. A., Swita, B., Romala, M., & Damanik, F. (2021). A Conjecture on Super Edge-Magic Total Labeling of 4-Cycle Books. International Journal of Mathematics and Mathematical Sciences, 2021, 1-8. https://doi.org/10.1155/2021/8483926

Simanjuntak, R., Bertault, F., & Miller, M. (2000). Two new (a, d)-antimagic graph labelings. Proc. of Eleventh Australasian Workshop on Combinatorial Algorithms, 11, 179–189, Two new (a, d)-antimagic graph labelings | Request PDF (researchgate.net).

Lakshmi, S., & Sagayakavitha, D. (2018). Edge-magic total labeling of some graphs. International Journal of Mathematics Trends and Technology (IJMTT), 1, 101–103, https://www.internationaljournalssrg.org/uploads/specialissuepdf/ICRMIT/2018/MTT/ICRMIT-P116.pdf.

Taimur, A., Numan, M., Ali, G., Mumtaz, A., & Semaničová-Feňovčíková, A. (2018). Super (a, d)-H-antimagic labeling of subdivided graphs. Open Mathematics, 16(1), 688–697. https://doi.org/10.1515/math-2018-0062

Ulfatimah, R., Roswitha, M., & Kusmayadi, T. A. (2017). H-Supermagic Labeling on Shrubs Graph and L_m⊙ P_n. Journal of Physics: Conference Series, 855(1). https://doi.org/10.1088/1742-6596/855/1/012055

Zeen El Deen, M. R. (2019). Edge even graceful labeling of some graphs. Journal of the Egyptian Mathematical Society, 27(1). https://doi.org/10.1186/s42787-019-0025-x




DOI: https://doi.org/10.31764/jtam.v8i2.20065

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