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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
A.N.M. Salman
Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia
Electrical & Electronics Engineering

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On the inverse graph of a finite group and its rainbow connection number Rian Febrian Umbara; A.N.M. Salman; Pritta Etriana Putri
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 11, No 1 (2023): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2023.11.1.11

Abstract

A rainbow path in an edge-colored graph G is a path that every two edges have different colors. The minimum number of colors needed to color the edges of G such that every two distinct vertices are connected by a rainbow path is called the rainbow connection number of G. Let (Γ, *) be a finite group with TΓ = {t ∈ Γ|t ≠ t−1}. The inverse graph of Γ, denoted by IG(Γ), is a graph whose vertex set is Γ and two distinct vertices, u and v, are adjacent if u * v ∈ TΓ or v * u ∈ TΓ. In this paper, we determine the necessary and sufficient conditions for the inverse graph of a finite group to be connected. We show that the inverse graph of a finite group is connected if and only if the group has a set of generators whose all elements are non-self-invertible. We also determine the rainbow connection numbers of the inverse graphs of finite groups.
Co-Authors Pritta Etriana Putri Rian Febrian Umbara