The Locating-Chromatic Number of Disjoint Union of Cycles
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Keywords:
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Color Code, Locating Chromatic Number, Cycle Graph
Abstract
Chartrand et al. introduced the idea of the locating-chromatic number of connected graphs in 2002. Let c be a disconnected graph H with k-coloring. Let S_i be the set of all vertices that get color i and let Phi be the partition of V(H) induced by c. The color code C_Phi(v)=(d(v,S_1), d(v,S_2), ..., d(v,S_k)) of a vertex v, where d(v,S_k)=min{d(v,x)} . The locating k-coloring of H is denoted by c if all vertices in H have unique distinct color codes. Welyyanti et al. in 2014 expanded on this idea so that it also applies to unconnected graphs. In this work, for n=>3 and m=>2, we calculate the locating-chromatic number of the disjoint union of cycles, represented by mC_n.
References
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Asmiati, H. Assiyatun, and E. Baskoro (2011). Locating-Chromatic Number of Amalgamation of Stars. ITB Journal of Science, 43A(1); 1–8
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Asmiati, E. Baskoro, H. Assiyatun, D. Suprijanto, R. Simanjuntak, and S. Uttunggadewa (2012). The Locating-Chromatic Number of Firecracker Graphs. Far East Journal of Mathematical Sciences (FJMS), 63(1); 11–23
Asmiati, W. Okzarima, Notiragayu, and L. Zakaria (2024). Upper Bounds of the Locating Chromatic Numbers of Shadow Cycle. International Journal of Mathematics and Computer Science, 19; 239–248
Asmiati, K. Prawinastia, M. Damayanti, and L. Yulianti (2025). The Locating Chromatic Number of (k, n)-Split Cycle Graph and Its Barbell Operation. Electronic Journal of Graph Theory and Applications, 13(2); 271–280
Asmiati, I. K. Sadha Gunce Yana, and L. Yulianti (2018). On the Locating Chromatic Number of Certain Barbell Graphs. International Journal of Mathematics and Mathematical Sciences, 2018(1); 5327504
Asmiati, A., D. Maharani, and Y. Lyra (2021). On the Locating Chromatic Number of Barbell Shadow Path Graphs. Indonesian Journal of Combinatorics, 5(2); 82–93
Assiyatun, H., D. K. Syofyan, and E. T. Baskoro (2020). Locating-Chromatic Number of the Edge-Amalgamation of Trees. Indonesian Journal of Combinatorics, 4(2); 125–131
Baskoro, E. T. and Arfin (2021). All Unicyclic Graphs of Order n with Locating Chromatic Number n − 3. Indonesian Journal of Combinatorics, 5(2); 73–81
Chartrand, G., D. Erwin, M. Henning, P. Slater, and P. Zhang (2003). Graphs of Order n with Locating-Chromatic Number n − 1. Discrete Mathematics, 269(1–3); 65–79
Chartrand, G., D. Erwin, M. A. Henning, P. J. Slater, and P. Zhang (2002). The Locating-Chromatic Number of a Graph. Bull. Inst. Combin. Appl, 36(89); 101
Damayanti, M., Asmiati, Fitriani, M. Ansori, and A. Faradilla (2021). The Locating Chromatic Number of Some Modified Path with Cycle Having Locating Number Four. In Journal of Physics: Conference Series, volume 1751. IOP Publishing, page 012008
Ghanem, M., H. Al-Ezeh, and A. Dabbour (2019). Locating Chromatic Number of Powers of Paths and Cycles. Symmetry, 11(3); 389
Hamzah, A., Asmiati, and D. W. Amansyah (2024). Locating Chromatic Number for Corona Operation of Path Pn and Cycle Cm, (m = 3, 4). Indonesian Journal of Combinatorics, 8(2); 127–135
Hamzah, N., Asmiati, and A. Nuryaman (2025). The Locating Chromatic Number for Corona Operation of Path and Cycle with Python Programming. Journal of the Indonesian Mathematical Society, 31(4); 1–9
Irawan, A., A. Asmiati, L. Zakaria, and K. Muludi (2021). The Locating-Chromatic Number of Origami Graphs. Algorithms, 14(6); 167
Sakri, R. and M. Abbas (2024). The Locating Chromatic Number of Generalized Petersen Graphs with Small Order. Examples and Counterexamples, 5; 100141
Sakri, R. and B. Slimi (2025). The Bound on the Locating-Chromatic Number for a Generalized Petersen Graphs P (N, 2). Examples and Counterexamples, 7; 100183
Sudarsana, I. W., F. Susanto, and S. Musdalifah (2022). The Locating Chromatic Number for m-Shadow of a Connected Graph. Electronic Journal of Graph Theory and Applications, 10(2); 589–601
Welyyanti, D., L. Abel, and L. Yulianti (2025). The Locating Chromatic Number of Chain(A, 4, n) Graph. BAREKENG: Jurnal Ilmu Matematika dan Terapan, 19(1); 353–360
Welyyanti, D., M. R. Fajri, I. M. Arnawa, I. P. Sandy, W. Arifitriana, L. A. Abel, F. Abbdurrahman, and N. Ghanny (2026a). Locating Chromatic Number of Small Circulant Graph. In AIP Conference Proceedings, volume 3389. AIP Publishing LLC, page 020012
Welyyanti, D., R. S. Zahra, and L. Yulianti (2026b). On the Locating-Chromatic Number of the Sunflower Graph. Journal of the Indonesian Mathematical Society, 32(1); 1–12
Arawan, A. and A. Istiani (2024). On the Locating-Chromatic Number of the Sunflower Graph. Sainmatika: Jurnal Ilmiah Matematika dan Ilmu Pengetahuan Alam, 21(1); 89–96
Arfin (2025). The Locating-Chromatic Number of Some Jellyfish Graphs. Journal of the Indonesian Mathematical Society, 31(1); 1–10
Asmiati, H. Assiyatun, and E. Baskoro (2011). Locating-Chromatic Number of Amalgamation of Stars. ITB Journal of Science, 43A(1); 1–8
Asmiati and E. Baskoro (2012). Characterizing All Graphs Containing Cycles with Locating-Chromatic Number 3. In AIP Conference Proceedings, volume 1451. pages 351–357
Asmiati, E. Baskoro, H. Assiyatun, D. Suprijanto, R. Simanjuntak, and S. Uttunggadewa (2012). The Locating-Chromatic Number of Firecracker Graphs. Far East Journal of Mathematical Sciences (FJMS), 63(1); 11–23
Asmiati, W. Okzarima, Notiragayu, and L. Zakaria (2024). Upper Bounds of the Locating Chromatic Numbers of Shadow Cycle. International Journal of Mathematics and Computer Science, 19; 239–248
Asmiati, K. Prawinastia, M. Damayanti, and L. Yulianti (2025). The Locating Chromatic Number of (k, n)-Split Cycle Graph and Its Barbell Operation. Electronic Journal of Graph Theory and Applications, 13(2); 271–280
Asmiati, I. K. Sadha Gunce Yana, and L. Yulianti (2018). On the Locating Chromatic Number of Certain Barbell Graphs. International Journal of Mathematics and Mathematical Sciences, 2018(1); 5327504
Asmiati, A., D. Maharani, and Y. Lyra (2021). On the Locating Chromatic Number of Barbell Shadow Path Graphs. Indonesian Journal of Combinatorics, 5(2); 82–93
Assiyatun, H., D. K. Syofyan, and E. T. Baskoro (2020). Locating-Chromatic Number of the Edge-Amalgamation of Trees. Indonesian Journal of Combinatorics, 4(2); 125–131
Baskoro, E. T. and Arfin (2021). All Unicyclic Graphs of Order n with Locating Chromatic Number n − 3. Indonesian Journal of Combinatorics, 5(2); 73–81
Chartrand, G., D. Erwin, M. Henning, P. Slater, and P. Zhang (2003). Graphs of Order n with Locating-Chromatic Number n − 1. Discrete Mathematics, 269(1–3); 65–79
Chartrand, G., D. Erwin, M. A. Henning, P. J. Slater, and P. Zhang (2002). The Locating-Chromatic Number of a Graph. Bull. Inst. Combin. Appl, 36(89); 101
Damayanti, M., Asmiati, Fitriani, M. Ansori, and A. Faradilla (2021). The Locating Chromatic Number of Some Modified Path with Cycle Having Locating Number Four. In Journal of Physics: Conference Series, volume 1751. IOP Publishing, page 012008
Ghanem, M., H. Al-Ezeh, and A. Dabbour (2019). Locating Chromatic Number of Powers of Paths and Cycles. Symmetry, 11(3); 389
Hamzah, A., Asmiati, and D. W. Amansyah (2024). Locating Chromatic Number for Corona Operation of Path Pn and Cycle Cm, (m = 3, 4). Indonesian Journal of Combinatorics, 8(2); 127–135
Hamzah, N., Asmiati, and A. Nuryaman (2025). The Locating Chromatic Number for Corona Operation of Path and Cycle with Python Programming. Journal of the Indonesian Mathematical Society, 31(4); 1–9
Irawan, A., A. Asmiati, L. Zakaria, and K. Muludi (2021). The Locating-Chromatic Number of Origami Graphs. Algorithms, 14(6); 167
Sakri, R. and M. Abbas (2024). The Locating Chromatic Number of Generalized Petersen Graphs with Small Order. Examples and Counterexamples, 5; 100141
Sakri, R. and B. Slimi (2025). The Bound on the Locating-Chromatic Number for a Generalized Petersen Graphs P (N, 2). Examples and Counterexamples, 7; 100183
Sudarsana, I. W., F. Susanto, and S. Musdalifah (2022). The Locating Chromatic Number for m-Shadow of a Connected Graph. Electronic Journal of Graph Theory and Applications, 10(2); 589–601
Welyyanti, D., L. Abel, and L. Yulianti (2025). The Locating Chromatic Number of Chain(A, 4, n) Graph. BAREKENG: Jurnal Ilmu Matematika dan Terapan, 19(1); 353–360
Welyyanti, D., M. R. Fajri, I. M. Arnawa, I. P. Sandy, W. Arifitriana, L. A. Abel, F. Abbdurrahman, and N. Ghanny (2026a). Locating Chromatic Number of Small Circulant Graph. In AIP Conference Proceedings, volume 3389. AIP Publishing LLC, page 020012
Welyyanti, D., R. S. Zahra, and L. Yulianti (2026b). On the Locating-Chromatic Number of the Sunflower Graph. Journal of the Indonesian Mathematical Society, 32(1); 1–12
Authors
Welyyanti, D., Fajri, M. R., Abel, L. A., Yulianti, L., Nurinsani, A., & Permana, D. . (2026). The Locating-Chromatic Number of Disjoint Union of Cycles. Science and Technology Indonesia, 11(3), 1046–1053. https://doi.org/10.26554/sti.2026.11.3.1046-1053
This work is licensed under a Creative Commons Attribution 4.0 International License.