آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 131 |
| تعداد مقالات | 1,631 |
| تعداد مشاهده مقاله | 1,879,815 |
| تعداد دریافت فایل اصل مقاله | 1,741,139 |
On the rainbow connection number of the connected inverse graph of a finite group | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 8، دوره 11، شماره 3، آذر 2026، صفحه 857-870 اصل مقاله (478.07 K) | ||
| نوع مقاله: Indonesia-Japan Conference on Discrete and Computational Geometry, Graphs, and Games | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29123.1848 | ||
| نویسندگان | ||
| Rian Febrian Umbara1، 2؛ A.N.M. Salman* 3؛ Pritta Etriana Putri3 | ||
| 1Doctoral Program in Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia | ||
| 2School of Computing, Telkom University, Bandung, Indonesia | ||
| 3Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia | ||
| چکیده | ||
| Let $\Gamma$ be a finite group with $T_\Gamma=\{t\in \Gamma \mid t\ne t^{-1} \}$. The inverse graph of $\Gamma$, denoted by $IG(\Gamma)$, is a graph whose vertex set is $\Gamma$ and two distinct vertices, $u$ and $v$, are adjacent if $u*v\in T_\Gamma$ or $v*u\in T_\Gamma$. In this paper, we study the rainbow connection number of the connected inverse graph of a finite group $\Gamma$, denoted by $rc(IG(\Gamma))$, and its relationship to the structure of $\Gamma$. We improve the upper bound for $rc(IG(\Gamma))$, where $\Gamma$ is a group of even order. We also show that for a finite group $\Gamma$ with a connected $IG(\Gamma)$, all self-invertible elements of $\Gamma$ is a product of $r$ non-self-invertible elements of $\Gamma$ for some $r\leq rc(IG(\Gamma))$. In particular, for a finite group $\Gamma$, if $rc(IG(\Gamma))=2$, then all self-invertible elements of $\Gamma$ is a product of two non-self-invertible elements of $\Gamma$. The rainbow connection numbers of some inverse graphs of direct products of finite groups are also observed. | ||
| کلیدواژهها | ||
| rainbow connection number؛ inverse graph؛ finite group | ||
| مراجع | ||
|
[1] M.R. Alfuraidan and Y.F. Zakariya, Inverse graphs associated with finite groups, Electron. J. Graph Theory Appl. 5 (2017), no. 1, 142–154. https://doi.org/10.5614/ejgta.2017.5.1.14
[2] S. Chakraborty, E. Fischer, A. Matsliah, and R. Yuster, Hardness and algorithms for rainbow connection, J. Comb. Optim. 21 (2011), no. 3, 330–347. https://doi.org/10.1007/s10878-009-9250-9
[3] G. Chartrand, G.L. Johns, K.A. McKeon, and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008), no. 1, 85–98. http://doi.org/10.21136/MB.2008.133947
[4] R. Diestel, Graph Theory, Springer, 2005.
[5] D.S. Dummit and R.M. Foote, Abstract Algebra, Wiley, 2003.
[6] O. Ejima, K.O. Aremu, and A. Audu, Energy of inverse graphs of dihedral and symmetric groups, J. Egyptian Math. Soc. 28 (2020), no. 1, Article number: 43. https://doi.org/10.1186/s42787-020-00101-8
[7] D. Fitriani and A.N.M. Salman, Rainbow connection number of amalgamation of some graphs, AKCE Int. J. Graphs Comb. 13 (2016), no. 1, 90–99. https://doi.org/10.1016/j.akcej.2016.03.004
[8] D. Fitriani, A.N.M. Salman, and Z.Y. Awanis, Rainbow connection number of comb product of graphs, Electron. J. Graph Theory Appl. 10 (2022), no. 2, 461–473. https://doi.org/10.5614/ejgta.2022.10.2.9
[9] H. Li, X. Li, and S. Liu, The (strong) rainbow connection numbers of Cayley graphs on Abelian groups, Comput. Math. Appl. 62 (2011), no. 11, 4082–4088. https://doi.org/10.1016/j.camwa.2011.09.056
[10] H. Li, X. Li, and Y. Sun, Rainbow connection number of graphs with diameter 3, Discuss. Math. Graph Theory 37 (2017), no. 1, 141–154. https://doi.org/10.7151/dmgt.1920
[11] X. Li and Y. Sun, Rainbow connection numbers of line graphs., Ars Combin. 100 (2011), 449–463.
[12] X. Li and Y. Sun, Rainbow Connections of Graphs, Springer New York, 2012.
[13] Y. Ma and Z. Lu, Rainbow connection numbers of Cayley graphs, J. Comb. Optim. 34 (2017), no. 1, 182–193. https://doi.org/10.1007/s10878-016-0052-6
[14] K. Mageshwaran, G. Kalaimurugan, B. Hammachukiattikul, V. Govindan, and I.N. Cangul, On $L(h, k)$-labeling index of inverse graphs associated with finite cyclic groups, J. Math. 2021 (2021), Article ID: 5583433. https://doi.org/10.1155/2021/5583433
[15] M. Murni, A.E. Hadi, I. Febry, and A. Abdussakir, Anti-adjacency and Laplacian spectra of inverse graph of group of integers modulo $n$, IOP Conf. Ser.: Mater. Sci. Eng. 807 (2020), Article ID: 012033. https://doi.org/10.1088/1757-899X/807/1/012033
[16] R.F. Umbara, A.N.M. Salman, and P.E. Putri, On the inverse graph of a finite group and its rainbow connection number, Electron. J. Graph Theory Appl. 11 (2023), no. 1, 135–147. https://doi.org/10.5614/ejgta.2023.11.1.11
[17] R.F. Umbara, A.N.M. Salman, and P.E. Putri, Rainbow connection numbers of the inverse graphs of some finite abelian groups, AIP Conf. Proc. 2480 (2023), no. 1, Article ID: 030003. | ||
|
آمار تعداد مشاهده مقاله: 484 تعداد دریافت فایل اصل مقاله: 500 |
||