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On the rainbow connection number of the connected inverse graph of a finite group | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 13 مهر 1403 | ||
| 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 | ||
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[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. | ||
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آمار تعداد مشاهده مقاله: 11 |
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