On the Rainbow Mean Indexes of Caterpillars

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Let G be a simple connected graph and c an edge coloring with colors that are positive integers. Given a vertex v of G, we define its chromatic mean, denoted by cm(v), as the average of the colors of the incident edges. If cm(v) is an integer for each v ∈ V (G) and distinct vertices have distinct chromatic means, then c is called a rainbow mean coloring. The maximum chromatic mean of a vertex in the coloring c is called the rainbow mean index of c and is denoted by rm(c). On the other hand, the rainbow mean index of G, denoted by rm(G), is the minimum value of rm(c) among all rainbow mean colorings c of G. In this paper, we determine the rainbow mean indexes of families of caterpillars, including brooms, and double brooms.