<![CDATA[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 .]]>
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