Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 3 2026 09 01 On the rainbow connection number of the connected inverse graph of a finite group 857 870 14850 10.22049/cco.2024.29123.1848 EN Rian Febrian Umbara Doctoral Program in Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia School of Computing, Telkom University, Bandung, Indonesia 0000-0001-5950-5300 A.N.M. Salman Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia Pritta Etriana Putri Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia Journal Article 2023 10 31 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. https://comb-opt.azaruniv.ac.ir/article_14850_c728ded1d0f0f7ee5ac55156c06605e5.pdf