Bilangan Terhubung Pelangi pada Graf Tengah (M(G)) dari Graf Ulat (C_(m,2))

Fuji Fauzia Kiayi (Program Studi Matematika, Universitas Negeri Gorontalo, Bone Bolango 96554, Indonesia)
Sumarno Ismail (Program Studi Pendidikan Matematika, Universitas Negeri Gorontalo, Bone Bolango 96554, Indonesia)
Nisky Imansyah Yahya (Program Studi Matematika, Universitas Negeri Gorontalo, Bone Bolango 96554, Indonesia)
Lailany Yahya (Program Studi Matematika, Universitas Negeri Gorontalo, Bone Bolango 96554, Indonesia)
Salmun K. Nasib (Program Studi Statistika, Universitas Negeri Gorontalo, Bone Bolango 96554, Indonesia)

Research in the Mathematical and Natural Sciences
Vol. 4 No. 1 (2025): November 2024-April 2025

Edge coloring of a graph is considered rainbow connected if the graph is connected and a rainbow path exists for every pair of points. The rainbow connection number of a graph, denoted as , represents the smallest number of colors required to make the graph is rainbow connected. This study examines the rainbow connection number of the middle graph of a caterpillar graph. The middle graph is a modified result of a graph , denoted as . It is described as a graph constructed from the intersection of a set of points and edges. The set of points in the middle graph consists of the combination of points and edges of the graph . Two points are considered adjacent if only they are connected in , or if one point corresponds to a point and the other corresponds to an edge adjacent to it. A caterpillar graph denoted by is a tree that will be a path if all the leaf points are deleted. The results of this research show the rainbow-connected number theorem for the middle graph of the caterpillar graph for .